Yeah the MM experiment (Michelson-Morley). It proved that the speed of light you measure is independent of your frame of reference, whether your standing still or moving relative to a light source, you will measure the speed of light to be c.
So what does this have to do with aether? Well aether is a theoretical substance through EM waves can propogate, which is considered to be 100% "stationary". With aether, since light/EM waves travel through it, the speed of light relative to aether must be fixed at c. However, the speed of light we measure all the time (regardless of our motion) is c as well (proven by MM, and a few other observations like sending atomic clocks on planes around the world). There is a problem here. Since our speed relative to light is c and the speed of aether relative to light is c, we must always be stationary with respect to aether! So if we're always at rest with respect to aether, it may as well not exist. You can't run through the aether, because as you walk around aether moves with you. This also means different people "see" aether differently. If we have 2 guys walking in different directions to each other, and aether is stationary relative to both of them you know there's something dodgy. Whether or not you want to accept this type of aether is up to you, but I personally don't.
As well as the MM experiment supporting special relativity (and various others), Maxwell's laws of EM waves (developed before MM & Einstein) goes hand in hand with special relativity, since from his equations it can be mathematically derived (long & complicated) that the speed of light is related only to the permittivity & permaebility of the medium it travels through, which have been proven to be constant (independent of your actual motion through it). Since special relativity is incompatible with the aether theory, it's almost certain that aether doesn't exist.
Of course we can't be
100% sure of anything (could be a worldwide conspiracy
) , but I'm inclined to believe in special relativity and no aether.