The level of doping is extremely high in degenerate semiconductors. I think, this will result in band gap narrowing in most of the cases. So, optical absorption spectroscopy to detect the bandgap absorption edge may be used. The required level of doping to achieve degenerate stage is different for different semiconductors, so it might be a good idea to check with theory in parallel.
In the case of degenerate semiconductors a strong blur of excitonic bands in the photoluminescence and reflection spectra occurs that, in particular, is due to the screening of Coulomb interaction between electrons and holes, bound in excitons. A similar effect is observed for excitonic absorption edge (edge steepness decreases).