My question is: what are the prospects for the pseudo-simultaneous detection of gravitational waves and neutrinos from the same supernovae? What sort of stuff could we learn from such an event, both about supernovae and neutrinos? In particular, what are the prospects for estimating the neutrino mass?
This article basically seems to answer the question. They quote from an earlier study:
"Although no CCSNe have currently been detected by gravitational-wave detectors, previous studies indicate that an advanced detector network may be sensitive to these sources out to the Large Magellanic Cloud (LMC). A CCSN would be an ideal multi-messenger source for aLIGO and AdV, as neutrino and electromagnetic counterparts to the signal would be expected. The gravitational waves are emitted from deep inside the core of CCSNe, which may allow astrophysical parameters, such as the equation of state (EOS), to be measured from the reconstruction of the gravitational-wave signal."
Since we know from SN1987A that neutrinos from a supernova can be detected at that range, that seems to be a "yes". The biggest uncertainty seems to be how much gravitational wave energy would be emitted by the supernova, and at what frequencies, which depends on a relatively detailed understanding of exactly how the matter moves around in the explosion, one simulation of which is illustrated in the (rather awesome) video in the article.