Which experiment failed to detect the medium light was assumed to travel through?

Michelson-Morley, in 1887. Light was understood as a wave, waves need something to wave in, so physics had posited the luminiferous ether filling all space. The Earth moves through it at 30 kilometres a second, so light sent along the direction of travel and across it should take measurably different times.

The apparatus was a stone slab floating in mercury so it could be rotated without vibration, splitting a light beam down two perpendicular arms and recombining it to look for a shift in the interference fringes. The precision was more than enough. The shift was absent. They repeated it at different times of year in case the Earth happened to be at rest with the ether at one moment, and it stayed absent.

It is often called the most famous failed experiment in history, and the description undersells it - a null result of that quality is a discovery. Attempts to save the ether produced the Lorentz contraction as a fudge; Einstein took the constancy of light speed as a starting assumption instead and rebuilt space and time around it. He was oddly vague afterwards about how much the experiment had influenced him, and historians still argue about it.