  We consider resonant neutrino conversions in magnetised matter, such as a
degenerate electron gas. We show how magnetisation effects caused by axial
vector interactions of neutrinos with the charged leptons in the medium can
induce a new type of resonant neutrino conversion which may occur even in
situations where the MSW effect does not occur, such as the case of degenerate
or inverted neutrino mass spectra. Our new resonance may simultaneously affect
anti-neutrino $\bar{\nu_a} \leftrightarrow \bar{\nu}_b$ as well as neutrino
$\nu_{a} \leftrightarrow \nu_b$ flavour conversions, and therefore it may
substantially affect supernova neutrino energy spectra. Using SN1987A data we
conclude that only laboratory experiments with long baseline such as ICARUS or
MINOS are likely to find neutrino oscillations due to their sensitivity to
small $\Delta m^2$. We also comment on the possibility of resonant conversions
induced by Majorana neutrino transition moments and mention the case of sterile
neutrinos $\nu_s$.
