Is information about the speed of light hidden in its spectrum? Can the speed of light in the vacuum (c) be inferred from the spectrum of light?
If that is not the case is it possible to tell from lights spectrum that it has entered a different medium,  e.g. can the correct fraction of c be inferred from the spectrum then? 
 A: This is not the case. The spectrum of light refers to the frequency content of the oscillations of light at any given point: you select a point, look at the electric field oscillations there, and decompose it into a superposition of waves of different frequencies. Thus the analysis is local, and the spectrum is a property of the source of light, and does not typically change upon transmission through a medium (unless, of course, there is absorption).
There is also nothing in the spectrum of a given source that will let you infer the speed of light in vacuum. Different sources have different spectra, but their light will always travel at speed $c$ in the vacuum.
On the other hand, light of different frequencies will experience different phase shifts upon transmission through a dispersive medium with a different refraction index for each frequency. This is not directly testable in an experiment as spectrometers only measure the intensity at each frequency, but you can retrieve the phase shift information in an interference scheme.
Thus, you need to match your phase-shifted light with an unperturbed copy, and measure the intensity. 

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If instead of simple detectors for the output rays you put in spectrometers (for an exceptionally well aligned interferometer) then you can read off the phase shift from the interference patterns in the spectra. This then gives you information about the refractive index, and from that you can infer the speed of light in the medium at a given frequency. (Note, though, that this is in terms of $c$, which is always assumed known.)
