How can I show that the speed of light in vacuum is the same in all reference frames? I have regularly heard that the Michelson-Morley experiment demonstrates that the speed of light is constant in all reference frames.
By doing some research I have found that it actually demonstrated that the luminiferous aether probably didn't exist and that the speed of light didn't vary depending on which direction the planet was travelling in. I don't see how it demonstrated that motion towards a light source for instance doesn't affect the observer's speed relative to the light, as there were no moving parts in the experiment.
The other sources I've looked at which say that the Michelson Morley experiment proved nothing like this one: Is the second postulate of Einstein's special relativity an axiom? and this one: How can we show that the speed of light is really constant in all reference frames? tend to say that Maxwell's equations were actually more significant to Einstein as they predict that light moves at a constant velocity, and this velocity has to be relative to something (or in relativity's case, everything). That something was thought to be the aether, but in the absence of that why could it not be relative to whatever emitted it? It seems like a more obvious immediate conclusion to come to than the idea that it's the same relative to everyone and all the counterintuitive results that ensue.
Another idea is that the speed of light is the universal speed limit and therefore must have a fixed value just to work under galilean relativity. 
But then that argument goes in circles:
"Why can't you go faster than the speed of light?"
"Because otherwise your mass becomes infinite."
"Why does your mass become infinite?"
"Because of Einstein's special relativity."
But this is based on the original fact that you can't go faster than the speed of light, so there's no argument I can find which completely answers why the speed of light has to be constant, other than that it has been regularly tested since.
So my questions are:


*

*Is there something I'm missing about the Michelson-Morley experiment or Maxwell's equations which explains my objections and definitively shows that the speed of light is constant and it is impossible to go faster than it?

*If not, is there any other specific example, ideally which would have been there for Einstein, which I can use to explain to people with no knowledge of relativity why it is the case?
 A: 
I don't see how it demonstrated that motion towards a light source for instance doesn't affect the observer's speed relative to the light, as there were no moving parts in the experiment.

Given that the earth is moving through space - and that the experimental fixture was at rest with respect to the earth - it follows that the experimental fixture was in in motion with respect to the vacuum of space. And if that's the case, the experiment should have measured different speeds of light in different directions. Instead, however, it measured an isotropic speed of light.
Physicists interpreted this "null" result as evidence that the speed of light is independent of the motion of the observer. That is, observers always measure an isotropic speed of light - even when in motion.  
That's the origin of the "principle of the constancy of the velocity of light".
If you are not spooked by alternative interpretations, you might find the following link interesting: link.
A: For a basic treatment of the Michelson-Morley experiment please see 1. It's not important to know the technical details of the experiment to answer your questions though. The only relevant thing is the result, let me put it in basic terms since you seem to struggle with the "physics slang": 
While the total velocity of a ball thrown from a truck is the sum of the velocity of the ball relative to the truck and the velocity of the truck relative to the observer, the velocity of a light beam emitted from the truck is not. Much more the velocity of the light beam seems completely independent of the velocity of the truck. 
Michelson and Morely didn't have a truck, they had the earth orbiting the sun.
Please make it clear to yourself that this experimental fact can be explained by stating that the speed of light is constant. If I say to you the speed of light is constant in every frame of reference, then the above result isn't surprising at all to you. 
But you want more. You want me to prove to you that the speed of light is universally constant. I cannot. There will never be an experiment that shows that this axiom is universally true. How should one ever construct such an experiment, how should one, for example, test the theory in the Andromeda galaxy? It's impossible, but it doesn't matter: Why not just stick with the axiom, as long as we can explain everything we see around us with it? 
As you already said there's an interesting connection between the 
invariance of the speed of light and Maxwell's equations. One can indeed prove that the speed of light has to be constant, otherwise, Maxwell's theory can't be true for all inertial frames. But this is no proof that can convince you either, since accepting Maxwells equations is no different to accepting the invariance of the speed of light. Furthermore, the basis of Einstein's theory is not the invariance of the speed of light, but the invariance of the speed of action. Which cannot be concluded from Maxwell's theory, even though it's a reasonable guess.
Physical theories are not provable. But as long as they comply with reality, we accept them as truths. 
Addendum: I recommend this short lecture for layman by R. Feynman on the topic. Feynman and I present a very similar line of reasoning.
A: As a stand-alone fact, the fact that nothing can go faster than the speed of light would be extremely hard to prove (and is not proved by Michelson-Morley experiment). But that fact is built into the Lorentz transformation, one of the key tenets of special relativity. In fact the Lorentz transformation is so fundamental that one can think of it as part of the “operating system” on which much of modern physics is built - if it wasn’t a fundamental law of nature, almost all of physics as we know it simply would not exist. And this is a vast body of knowledge whose validity has not only been demonstrated experimentally in countless experiments, but is essential to the functioning of many technological devices, including some that we see all around us in our daily lives. So the fact that all these technological devices work as they are supposed to provides ample (if indirect) evidence that the speed of light is the universal speed limit.
A couple of specific examples you can read about in various places (e.g., in this article) are old fashioned TVs based on cathode-ray tubes, and speed-measuring radar guns used by the police, both of which inventions need to take relativity into account in their design. For example, in CRT televisions, a beam of high-velocity electrons is deflected by a magnetic field to hit a specific point on the TV screen, causing that point to light up. These electrons are moving at a fairly sizable fraction of the speed of light (20-30 percent according to the article I linked to), so relativity predicts you have to “push” them with a stronger force in order to deflect them by the right amount than you would under Newtonian mechanics. This is an example of how a body’s inertial mass grows as its velocity increases, according to a formula which would cause the mass to approach infinity as the body’s speed approaches the speed of light. So, if you are willing to believe that one can extrapolate from this formula as it applies to objects moving at a quarter of the speed of light to things going at 99% of the speed of light or even faster, you should agree that CRT televisions give good “everyday life” evidence that nothing can go faster than light.
A: The issue is more subtle than that, and the way you've phrased it is technically wrong. There are "reference frames" you can create in which light, or photons, travel with a different speed. This came up here:
(hmm, can't find link atm)
referencing the case of thinking about the viewpoint of a spinning person - the stars at great distances will seem to move "faster than light" thanks to $v = r\omega$, and so too will the photons, i.e. "light speed" will be higher, in fact, at suitable distance, arbitrarily high. (E.g. if you spin at an $\omega$ of 1 rad/s, and $r$ is even $1\ \mathrm{Pm}$, i.e. only 1/40th the distance to the nearest star away from the Sun, already the $v$ is $1\ \mathrm{Pm}/s$, far in excess of $c$, i.e. $3 \times 10^{-7}\ \mathrm{Pm/s}$ on this scale.)
In special relativity it's not brought up, but it is crucial in general relativity, and since general relativity includes special relativity as a special case (hence the name), the same cosiderations, technically speaking, apply to it.
Relativity is really a theory of space and time, and as said, it "requires the language of events, not things" (ref). More particularly, relativity is a theory about the laws that govern flows of information throughout the Universe. In its purest form, we are really only concerned with one kind of question, and it is this:

"Can you send a message from event $A$ to event $B$?"

"Events" are just points in space-time, to which we attach a signifier. The answer to this question is either "yes" or "no", a binary answer. For every pair of events in space-time, we can ask such a question, and the theory of relativity provides a mathematical framework that describes when the answer is "yes" and when the answer is "no". It also lets us work out how things look from the viewpoint of being inside a universe where the information we receive is subject to these constraints, i.e. what we can and cannot gather from the information coming at us at the little points in space-time we occupy. All the "weirdness" of relativity traces to basically this. Special relativity describes the form of those relations in the absence of matter, while general relativity describes how they are altered by the presence of matter.
Coordinate systems, or "reference frames", are simply ways to label events. What labels you put on them do not change the relationships between them. If I label the inside of my house "cooties", and the outside "znabby", that doesn't change the basic relationship of interiority/exteriority that exists between them any more than if I label them "inside" and "outside", respectively. (Same if I decide to confusingly call the outside "inside" and the inside "outside".)
What the Michelson-Morley experiment shows is not directly a statement about what happens in reference frames, or a statement about "aether", even - it is entirely logically possible to imagine a Minkowskian space-time filled up with an aetheric medium just as one can imagine a Galilean one so filled. Rather, it is a demonstration that the behavior of communications - of messages - obeys the former set of flow rules, not the latter.
And those rules can essentially be described as saying there exist a class of reference frames (coordinates, labels) that you can put on events, such that the permissibility of communication takes the form of a speed limit, and the transformation between these frames leaves said speed limit fixed. The reference frames follow from the limits, not that the limits follow from the reference frames. And one more result of the experiment is that it shows us that light, specifically, is a real-life medium of communication that saturates the Universal speed limit (to at least the experiment's error bounds, of course). 
When you relabel those with something else, like a rotating reference frame, of course these relationships become harder to describe mathematically, but they are still the same in that in both frames you will note that communication between the same sets of events is or isn't impossible. E.g. while light may be going "faster than light" in the rotating frame, you won't see any ships making a trip from Earth to Proxima b in less time than 4.3 years (barring of course potential things like wormholes that require GR and even more, still-unknown post-GR physics to fully treat and also to assess the (im)possibility of).
Now, as to why the communication rules in our Universe take this form, there really isn't an answer, at least that you can have in physics and to the best of our knowledge. The only way you can answer "why" in physics is if you can derive it, as you are suggesting, from a more fundamental principle, and your circular argument shows you can't, and moreover, when phrased as above it seems pretty damn fundamental already, so I would doubt that we will ever find such a reason. You have to start somewhere.
The best way to convey it is to just say that that our totality of empirical observation has been consistent with the idea that the Universe has a speed limit, and no exceptions have been found. That's it; it's "how it was made".
A: It's hard to fully understand what you're asking but here are some things that might help:


*

*The Michelson-Morley (MM) experiments don't show the speed of light is constant, it just rules out particular kinds of ether (the kind that can freely flow past particles). Ether is the supposed thing that light waves "oscillate" in. 

*You are right to say that's circular reasoning.

*Maxwell's equations don't prove the speed of light is constant. But they suggest it if you also assume it's not possible to tell which frame you are in.

*Einstein came to his conclusions based on gut instinct that electromagnetism had to obey the principle of relativity. He took this one step further and decided to elevate the idea to a principle and see what that lead to.

*There is no proof the speed of light is constant except experiment - you can't do it theoretically. There are some arguments that came after Einstein, based on the idea of causality, etc.

*Remember Einstein did physics by having convictions about the way the world worked and this worked exceptionally well for relativity but not for quantum non-locality.
A: 
That something was thought to be the aether, but in the absence of that why could it not be relative to whatever emitted it?

This seems to be the key point of your question.
Now, historically speaking this alternative had already been ruled out on theoretical grounds, because it isn't compatible with Maxwell's equations.  However, since you are asking for experimental evidence, consider astronomical observations of binary star systems: it seems obvious that if the speed of the light coming from a star in the part of its orbit when it is approaching us were different to the speed of light coming from the same star in the part of its orbit when it is going away from us, it would cause observable effects.
In order to help quantify the extent to which making the speed of light relative to the speed of the light source would affect astronomical observations, I did a Google search on the phrase "astronomical observations binary star systems speed of light" which found this article by Tedd Bunn, chair of the department of physics at the University of Richmond.  The answer is that the effect would be extremely obvious:

[...] you might wonder whether this effect would be significant for real star systems. The answer is that it turns out to be extremely significant. The reason is that the stars are very far away. That means that, even though the difference in the light's speed at the top and bottom of the orbit would be very slight, the faster light would still have lots of time to overtake the slower light. In fact, in real star systems, you'd end up seeing not just three or five images of the star, but thousands of images.
Needless to say, we don't see that sort of thing at all, and that provides extremely strong evidence that the speed of light does not depend on whether the source is moving towards you or away from you.

A: There are several issues with claiming empirical "proof" of the speed of light being constant:


*

*The problem of induction: MM tested several different conditions, and the result of identical travel times was inconsistent with several hypotheses involving differing speeds of light. However, there's no way through experiment to conclusively rule out all hypotheses. All we can say is that at this point, Occam's Razor strongly support the constancy of the speed of light.

*The impossibility of measuring the speed of light from one point to another: It is impossible to measure the time that it takes for light to go from one point to another. The reason for this is that to measure the speed of light from point A to point B, you need to know the time at which the light left point A and the time it arrived at point B, which means that you need to sync up your clocks at the two points.  Doing so requires sending a signal from one point to another with a timestamp, and then adjusting for travel time. But adjusting for travel time requires knowing the time it took, leading to circular reasoning. The only way to truly measure the speed of light is to measure the round trip time, which then gets you the average speed (for the appropriate meaning of "average"). If we assume that light travels at the same speed both away and towards us, then the speed in either direction is equal to the average speed. But it is completely consistent with empirical observations (albeit not with Occam's Razor) to establish a coordinate system in which light travels at a different speed depending on whether it's traveling toward us or away from us.

*"Reference frame" is a broad category: In the broadest sense, "reference frame" simply refers to a coordinate system. You probably mean "inertial reference frame". In non-inertial reference frame, the coordinate speed of light is not constant. Thus, to a large degree, "the speed of light is constant" is not a statement about the empirical world, but coordinate systems. Basically, it's "The speed of light is $c$ in any 'reasonable' coordinate system."
If we put aside those objections, then MM was reasonably conclusive.

there were no moving parts in the experiment.

Well, it took place on Earth, and Earth is moving in the solar system's reference frame.

I don't see how it demonstrated that motion towards a light source for instance doesn't affect the observer's speed relative to the light

MM compared the speed of light when it was traveling perpendicular to the Earth's motion versus parallel. If the motion of the detector relative to the light affected the speed, then those two conditions would have returned different results. Imagine a river, and you compare rowing a boat across the river and back, versus rowing it downstream then back. The travel times for those two roundtrips will be different, because the boat's speed changes depending on whether it's traveling with, against, or perpendicular to the current. Similarly, if the speed of light is affected by whether it's traveling with, against, or parallel to the motion of the Earth, then MM would have found a difference in travel time.

That something was thought to be the aether, but in the absence of that why could it not be relative to whatever emitted it? 

Within the framework of Maxwell's equations, the propagation of light is due to  changing electric field causing a changing magnetic field causing a changing electric field, etc. Once you have that, the propagation speed follows from the constants in Maxwell's equations. For the speed of propagation to depend on the speed of the emitter would require that these constants somehow change based on the emitter's speed. 
A: In response to question #1, yes, there are things you are missing about the M-M experiment and what exactly it showed, and why. Many excellent and detailed descriptions of it are available; have you consulted any of them? Regarding Maxwell's derivation of 
$$c = \frac{1}{\sqrt{\eta \mu}},$$ he did not obtain $c$ = (that expression) $\pm$ (the velocity of the laboratory), which means that special relativity was woven into his equations in a way that had to wait on Einstein to definitively uncover.
I think that once you've grasped (#1), you'll be able to answer (#2), but I am willing to concede the point to one of the experts on this site.
A: It can be shown by experiment that when light is emitted from a moving source, its wavelength changes but its speed doesn't. The Andromeda galaxy is an example. To prove that changes in wavelength don't affect the speed of light, you could fire several different color lasers at the reflector which astronauts put on the moon. You should get a return pulse in about 2.5 secs (the time varies slightly depending on exactly where the moon is in its elliptical orbit). If all the colors return in exactly the same time, you will have proved that although a change of speed by the source will change the wavelength, a change of wavelength doesn't mean there is a change of speed.
