  It has been noted for a long time that, in some circumstances, {\sl massless}
neutrinos may be {\sl mixed} in the leptonic charged current. Conventional
neutrino oscillation searches in vacuum are insensitive to this mixing. We
discuss the effects of resonant massless-neutrino conversions in the dense
medium of a supernova. In particular, we show how the detected $\bar\nu_e$
energy spectra from SN1987a and the supernova $r$-process nucleosynthesis may
be used to provide very stringent constraints on the mixing of {\sl massless}
neutrinos.
