Using under-sampled data, what determinations can be made about the Nyquist rate (correct sampling rate)? If my instrumentation limits me to sampling data at a fixed but insufficient rate what, if anything, can be determined from that data about the sampling rate that would be required to properly represent the signal? Are there signal analysis techniques that would provide any indicators?
This may be a trivial answer, but you can of course use the aliasing information to determine the correct sampling rate, given that you know that the signal is undersampled. In the simplest case, given that the signal appears to be at $k f_s,\ k < 0.5$ one can devise that the frequency of the signal is $(1-k)f_s$. In general it may be at $(N \pm k)f_s$ for some integer $N$.
If the modulation is asymmetric (as in SSB/USB) and appears to be "upside-down", then that may be evidence that a better sample rate is higher.