It is said that they have not…
But it is also said that electromagnetic field propagates with the speed of light. If there is no medium what exactly is propagating? How can “nothing” propagate?
It’s a “field”. It doesn’t propagate in anything. And what we call “light” is just the visible (to us) part of the EM spectrum. Of course EM waves propagate at the speed of light-- they are light (of certain wavelengths).
But keep in mind that we shouldn’t say light is a wave. It acts like a wave (except when it doesn’t 
).
And if you’re wondering, “relative to what?”, it’s relative to everything. All observers, moving at any speed at all, will all measure the speed of a beam of light to be the same. This is the core of Einstein’s Special Theory of Relativity.
Photons, the force-carrying particle for electromagnetic radiation,
are not nothing, and they are what propogates. The “nothing” part
refers to the fact that photons need no medium to propogate, they
may do so in a vacuum.
For several decades after the discovery (by Maxwell) of the unity of
all forms of electromagnetic radiation physicists believed that a
medium was necessary for propagation. This medium was termed
“lumineferous ether”, and was thought to be ubiquitous.The problem
was, a series of highly sophisticated experiments turned up no evidence
of ether’s existence. Google “Michelson-Morley” on that note.
Special Relativity did not disprove the existence of ehter, but it did
prove that ether was an unnecessary construct. Consequently ether
has disappeared from the working vocabulary of physics.
I was always interested in science & physics but since the luminiferous ether had long since gone the way of things like Piltdown Man and the ‘Steady-State’ universe, when I first heard mention of it in my late teens it was so alien to me that for the first few moments I thought, “Wait, they used to think the universe is filled with *starting fluid?!? *Geez, nobody light a match!” 
Of course they propagate through a medium. That medium is electric and magnetic fields. The field is the medium for the waves. The field vibrates, which is what the wave is. Just like air vibrating is a sound wave.
Special Relativity is a theory and theories don’t really prove or disprove anything, except in Math, I guess, where they’re called theorems. Experiments do (OK they don’t either. They give us evidence on the likelihood of things.)
The Michelson-Morely experiment and others cast doubt on the ether idea and many other experiments were done to see if the Earth dragged the ether along with it, or gravity affected the ether, how to explain stellar aberration, etc.
As far as I know, ether hasn’t really been disproved, it’s just not needed so Occam’s razor rather than experiment says it doesn’t exist. If it has been disproved, I’d guess it was via ideas from quantum mechanics rather then relativity.
I guess this is as good a place as any to be a stickler for precise expression.
I should have hauled out conditional the verb tenses and said something like :
“Experimental confirmation of SR would not disprove ether.”
However, SR is largely deductive and I do think it is reasonable to say the logic
of the theory does disprove the necessity for ether’s existence.
I thought Blaise Pascal pretty convincingly proved the existence of a vacuum. I would think most of the arguments would end up there and have no place else to go. Yet hundreds of years later they were still struggling with the concept of ether.
Vacuum versus ether? I guess they thought the question wasn’t ether-or.
I have heard an alternative way of describing EM radiation.
A description where an electromagnetic field pervades all of space, and photons are basically ripples of this field (yes, it was in the context of explaining the Higgs field, and the claim was that the EM field can be modeled in the same way).
If this is true, and this model is equivalent, then the OP may find this more intuitive.
The OP’s question is a wholly reasonable one – if light and other EM radiation is a wave, what’s it a wave in? This question and the attempt to find an answer drove physics for decades. It’s not really correct to dismiss this so simply and flippantly as to say “it’s a wave in nothing”. The answer needs elaboration and context. There’s a vast literature on the topic. You can start, as is often the case on the internet, with Wikipedia.*
*I really should send these guys some money, or something.
I’m going off some pretty weak memory here but I seem to recall that Relativity when applied to the Michelson-Morley experiment meant that the experiment would not detect the ether even if it was there.
Or in other words M-M was designed to detected upstream/downstream/sidestream in the ether basically. And with a Newtonian point of view if the ether was there it would detect it. However, when you apply the Relativistic effects, the shrinkage/time dilation happens to perfectly cancel out the effects of going upstream/downstream/sidestream that the experiment is meant to detect.
Or is my memory crap and the coffee not fully kicked in yet?
A scientific theory is not the same sort of animal as a mathematical theory.
I don’t understand the problem here. Electromagnetic waves are ripples in electric and magnetic fields. The electric field wiggles, which causes a vibration in the magnetic field, which makes the electric field ripple, which causes the magnetic field to shimmy, and so on ad infinitum.
Electric and magnetic fields exist, so to say that EM waves are ripples “in nothing” is factually incorrect. Maybe you find it counterintuitive that atoms aren’t involved, but I don’t see why they have to be. Fields exist and can vibrate too.
No, that’s pretty much correct. If you try to analyze the M-M experiment in a reference frame where the device is moving, you’ll observe the arms to be contracted along the direction of motion, and you’ll also observe time to be dilated as measured by “clocks” moving along with the device. These two effects cancel out and mean that you don’t get any interference effects due to the “differing speed of light” in the various directions, as measured relative to the device.
Maybe the difficulty is in imagining what the waves and fields are made of. You can feel force field and see amber waves of grain, but what makes up EM waves that they can exist in the vacuum of space?
It’s more difficult to conceptualize because (if I have it right) the EM field in which EM waves propagate is essentially created by the propagation itself. This is as opposed to sound, which propagates through a medium that has a physical existence independent of the sound wave passing through it.
I don’t think that model would work with relativity - the field itself would provide an absolute reference frame - and someone travelling relative to it, but in the opposite direction to a set of ripples, would perceive those ripples as moving faster than c