Isn't LIGO basically measuring the luminiferous aether? I am bit confused about this one. I am not very acknowledgeable about gravitational waves and LIGO. But if it is basically a Michelson interferometer and can detect shifts in vacuum, doesn't this means that we detected the luminiferous aether and if not, why? Is there an analogy or direct correlation between gravitational waves and aether?
 A: 
Isn't LIGO basically measuring the luminiferous aether?

Not quite. Instead it's measuring "waves in the aether". Take a look at the Einstein digital papers, and note this:
"Recapitulating, we may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an aether". 
This isn't quite the same as the luminiferous aether, but it isn't totally different. Also see this quote by Robert B. Laughlin, Nobel Laureate in Physics, endowed chair in physics, Stanford University: 
"It is ironic that Einstein's most creative work, the general theory of relativity, should boil down to conceptualizing space as a medium when his original premise [in special relativity] was that no such medium existed [..] The word 'ether' has extremely negative connotations in theoretical physics because of its past association with opposition to relativity. This is unfortunate because, stripped of these connotations, it rather nicely captures the way most physicists actually think about the vacuum..."

I am bit confused about this one. I am not very acknowledgeable about gravitational waves and LIGO. But if it is basically a Michelson interferometer and can detect shifts in vacuum, doesn't this means that we detected the luminiferous aether and if not, why? 

Because whilst space is the aether of general relativity, it isn't actually the luminiferous aether. I imagine your question will be What's the difference between the luminferous aether and the aether of general relativity? I'm afraid to say I don't think I can give a good answer to that. But note that Einstein described space as the aether of general relativity, not spacetime. Spacetime is an abstract thing, akin to the "block universe". See what relativist Ben Crowell said here: "Objects don't move through spacetime. Objects move through space". 
A: How a Michelson Interferometer works
LIGO is indeed a Michelson interferometer, which splits a beam of light in two, sending each new beam towards a mirror in the shape of a giant "L". Mirrors at the end of each arm then reflect the light and send the beams back to the splitter, where they merge back. An interferometer looks to find any changes in the beams. If a gravitational wave passes through the detector, it will appear to change the length of the path traveled by a beam, thus changing the result slightly.1
The Michelson-Morley experiment did pretty much the same thing. Michelson's idea was that the apparent length should change based on the Earth's supposed movement through the aether. It would take longer for the beam traveling parallel to the Earth's movement through the aether to return than if would for the beam traveling perpendicular to the Earth's movement.
The difference between the experiments
If we detect a gravitational wave, then we can still rule out the luminiferous aether. There are a couple reasons:


*

*If the aether hypothesis is true, the movement of the Earth through the aether should always be detectable. Conditions stay the same. However, gravitational waves do not regularly travel through Earth at measurable strengths.

*In the Michelson and Morley setup, the Earth is traveling through the aether in one direction, and thus changing the orientation of the interferometer should produce different results. However, gravitational waves can come from any direction, so it is possible to get the same measurement from different waves coming from different directions.


This is, of course, in addition to the fact that spacetime is not the same thing as the aether.

1 For more excellent information, see the LIGO website.
A: Here is my less-than-scholarly answer.  Please tell me its weaknesses:  The luminiferous aether was postulated as a medium necessary to the transmission of light-waves. If the light-waves were merely a perturbation of a medium, then the motion of an observer through that medium would change the apparent speed of the perturbations through which the observer was traveling. The Michelson-Morley experiment proved that this wasn't the case.  It was the empirical proof of Einstein's premise that the speed of light appears the same for all observers in uniform motion with respect to one another, regardless of whether they are moving toward or away from the source of the light. Einstein's "aether" was not a medium, but rather represents the property of space-time that has the effect of "warping" the path of a moving object, including a light beam, in proximity to a massive body. Since the warping takes place without actually coming into contact with the body, it is somewhat analogous to the luminiferous aether in that it postulates the ability to have "action at a distance" (e.g. the "warping"), but with the luminiferous aeither, that action was a traveling perturbation through a substance.  Gravity waves represent oscillations in the warping caused by variations in the acceleration of the massive body relative to the observer.
A: You are right, spacetime is the luminiferous aether (that kind of aether that Einstein abhorred) and gravitational waves are ripples in that aether. But, the Michelson-Morley experiment showed us in 1887 that there is no luminiferous aether, therefore gravitational waves do not exist or they cannot be detected by means of a LIGO interferometer.
