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JoshuaZ comments on Particles break light-speed limit? - Less Wrong
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Yes, but we almost always measure c precisely using light near the visible spectrum. Rough estimates were first made based on the behavior of Jupiter and Saturn's moons (their eclipses occurred slightly too soon when the planets were near Earth and slightly too late when they were far from Earth).
Variants of a Foucault apparatus are still used and that's almost completely with visible light or near visible light.
One can also use microwaves to do clever stuff with cavity resonance. I'm not sure if there would be a noticeable energy difference.
The ideal thing would be to measure the speed of light for higher energy forms of light, like x-rays and gamma rays. But I'm not aware of any experiments that do that.
The experimental upper bound on photon mass is 10^-18 eV. The photons near visible spectrum have about 10^-3 eV, which means their relative deviation from c is of order 10^-30. Gamma would be even closer. I don't think mass of photon is measurable via speed of light.
Err... build a broad spectrum telescope and look at an unstable stellar entity?
That's an interesting idea. But the method one detects gamma rays or x-rays is very different than what one uses to detect light, so calibrating would be tough. And most unstable events take place over time, so this would be really tough. Look at for example a supernova- even the neutrino burst lasts on the order of tens of seconds. Telling whether the gamma rays arrived at just the right time or not would seem to be really tough. I'm not sure, would need to crunch the numbers. It certainly is an interesting idea.
Hmm, what about actively racing them? Same method as yours but closer in. Set off a fusion bomb (which we understand really well) far away (say around 30 or 40 AU out). That will be on the order of a few light hours which might be enough to see a difference if one knew then that everything had to start at the exact same time.
Short answer: The numbers come out in the ballpark of hours not seconds.
Being closer in relies on trusting your engineering competence to be able to calibrate your devices well. Do it based off interstellar events and you just need to go "Ok, this telescope went bleep at least a few minutes before that one" then start scribbling down math. I never trust my engineering over my physics.