Who discovered the speed of light is constant? As we know that Michelson & Morley's experiment was not able to detect any effect of Aether on the motion of light, but if we follow the classic mechanism of law of motions as peoples do at that time{i.e. even speed of light supposed to be relative}, then how does it lead to conclude that speed of light is constant for all the observer irrespective of any frame of references. So what lead Lorentz to introduce Lorentz-transformation.
[please add comments if I misunderstood something..]
 A: Note, the question asked is about history, and historically the definitive statement that the speed of light is independent of other things comes from Einstein's 1905 paper. In developing his thinking Einstein was unaware of the Michelson Morley experiment. To quote from https://arxiv.org/ftp/arxiv/papers/1204/1204.3390.pdf:
'When Robert Shankland asked Einstein how he had learned of the Michelson-Morley experiment, Einstein told him that he had become aware of it through the writings of Lorentz, but only after 1905 had it come to his attention. "Otherwise", he said, "I would have mentioned it in my paper".'
It is also not true to say that the Maxwell equations, in and of themselves, were able to resolve the issue. It is only with hindsight that we can see them as fully consistent with the idea of a universal speed of light, in which the constants $\epsilon_0$ and $\mu_0$ are independent of reference frame.
The experimental evidence helping to motivate Einstein's thinking was chiefly stellar aberration and Fizeau's experiments on the speed of light in moving water, which are very hard to square with aether-based models. It was this, combined with the Maxwell equations and Einstein's own penetrating physical reasoning, which led him to propose that the speed of light was the invariant quantity, and notions of simultaneity and physical distance were frame-dependent.
To summarize, the Michelson Morley experiment is rightly celebrated as an excellent experiment, but it was not the main evidence motivating the notion that the speed of light is a universal, and in fact it had no impact at all on the breakthrough piece of reasoning (by Einstein) concerning that fact. Rather, it came in as a further piece of evidence, and as an experimental method of exquisite sensitivity which would then be applicable to many other applications (eventually underpinning the gravitational wave detectors of today).
A: Before Michelson and Morley did their experiment, Maxwell had formulated his famous equations describing electromagnetism. The nature of light as a "freestanding" (within the assumed aether) electromagnetic wave was then deduced from these equations. Additionally, the equations imply that such a freestanding wave must have a specific speed, but this was initially interpreted as a speed compared to the aether, analogous to how still air has a uniform speed of sound.
The Lorentz transform (actually discovered by Poincare) transforms coordinates from one reference frame (i.e. choice of origin point and 3-velocity of an imaginary coordinate grid) to another which is moving at some constant velocity compared to the first and does so in such a way that Maxwell's (and incidentally, Newton's) equations are still found to be true in the second frame.
This is unlike, e.g. within an atmosphere, where all but one choice of velocity of your frame will result in objects spontaneously accelerating up to a certain velocity if left unsupported. This "certain velocity" is of course the velocity of your frame relative to the air, and the "acceleration" is atmospheric drag returning the object to being at rest with respect to the air. The one velocity where this doesn't happen is when you were at rest inn the air to begin with - but if you picked randomly, that would be unlikely.
I'm not sure exactly how the transform was viewed prior to MM proper, but we can imagine they're a mathematical curiosity at this point - these sets of transforms have this property, but no apparent physical application because the aether exists, and Maxwell's equations shouldn't directly apply in non-stationary frames just like Newtonian physics changes slightly in frames not stationary compared to an atmosphere
But remember that Maxwell's equations imply a particular speed for light. So obviously the same thing should happen - most frames would be moving through the aether at some non-zero velocity, any light they emit should experience aether drag just like physical objects experience atmospheric drag, and light should be (at least slightly) bent or slowed by that drag so that it maintains the speed Maxwell says it should in the aether's frame.
And then Michelson and Morley came along and that idea suddenly became very dicey. If you always measure the speed of light as Maxwell says you should, no matter what velocity you have, the equations are actually invariants, and must remain true even as your velocity changes. Which in turn means, physical measurements must change according to the Lorentz transformations. This is Relativity.
So to answer the title, Michelson and Morley. But the argument that got them from "no aether drag" to "speed of light is constant always" was built on the foundation of Maxwell's equations.
A: Very interesting question. At first I would like to note the theoretical foundations upon which Einstein built his special relativity postulates. His main influence was Maxwell's equations - from these can be extracted that the speed of light $c$ depends only on vacuum permittivity & vacuum permeability :
$$ c={\frac {1}{\sqrt {\varepsilon _{0}\mu _{0}}}}$$
As you can see, no terms dependent on any kind of speed, or ether properties or whatever. A short and beautiful equation.
As for experimental confirmations. I will not argue about who was the first who showed light speed invariance, because there are some serious issues with experimentation. Probably first should be Michelson and Morley, but :

*

*The main conclusion which can be drawn from their experiment is that there doesn't exist an ether which drags light with it, thus changing relative light speed. But this depends on ones understanding of "ether" and it's properties, and what you try to look for. There could still exist, given their result, a preferred reference frame which still can "drag light" with it, but one that we just don't know how to search for. The point is that it is very hard to disprove something if you don't know what to look for. Null result experiments (of which, M&M, is one) can be interpreted in many ways.


*That, and many subsequent experiments, have concentrated on measuring light speed invariance in "two-way light speed" setups. This means that the experimenter is measuring round-trip light speed - light going from a source to a detector and back again. As a result the experimenter doesn't need to think about how to synchronize the clocks at the source and detector. Most light speed invariance experiments deal with this type of test.
However trying to measure light speed invariance in a one-way light speed setup is ongoing research with inconclusive results. For example, Stephan J. G., and others has found some light speed anisotropy - a light signal going around Earth at the equator takes $207~\text{ns}$ longer when it goes in an eastward direction as opposed to west. They explain this result as $c \pm v$ light speed relative variability due to Earths rotation. But I have not seen this in their setup; did they try to measure light speed in other perpendicular directions - light going around Earth north vs. south, and find they were the same ? If they would get similar results in a speed difference - then this means that outcome has nothing to do with Earth rotation.
Additional criticism of such light speed anisotropy experiments is that often researchers assume that anisotropy is mutually incompatible with special relativity, which may be not the case. Einstein did not say that there can't exist a preferred reference frame. He said only that in inertial reference frames light speed should be invariant. Ultimately one should understand that Earth is a non-inertial reference frame, because everything is subject to gravitational force, which according to Einstein's equivalence principle is similar to a reference frame moving with an acceleration. So one needs to be very careful in not mixing special and general relativity theories. One performing such light speed anisotropy experiments needs to very carefully think on a) how to synchronize clocks at source and detector and b) how to eliminate general relativity's effects on experimental setup and calculations.
A: There are experiments that were used to disprove a rival theory to relativity called Emmission Theory. The rival theory posited that the speed that light travels is equal to $c + v$, where $c$ is the velocity of light from a stationary source and $v$ is the velocity of the light emitter. Observations of stars orbiting each other disproved this theory when scientists noted that, if Emission Theory were true, then light emitted from a star traveling towards Earth would be able to overtake like emitted earlier from the star that was moving away. This would result in a scrambled appearance of the binary star due to light arriving out of order. So, light emitted from an approaching source travels at the same speed as a receding source.
