  It is known that if neutrino is a Dirac fermion with magnetic moment, then
\nu_L -> \nu_R -> \nu_L transition of supernova neutrinos may occur. The first
stage of such transition is due to the neutrino spin flip inside the hot dense
supernova core, while the second one - due to the neutrino spin precession in
the galactic magnetic field on the way from the supernova to terrestrial
detectors. This can result in the detection of 60-200 MeV neutrinos
simultaneously with the "normal" supernova neutrino signal, which would be a
smoking gun for the Dirac neutrino magnetic moment, \mu. We argue that in case
of a nearby supernova explosion (~10 kpc away from the Earth) one may observe
such high-energy events in Super-Kamiokande if \mu \gtrsim 10^{-13} \mu_B, and
in a Mt-scale detector if \mu \gtrsim 0.5*10^{-13} \mu_B. Such an observation
by itself, however, may be not sufficient to determine the value of the
magnetic moment, because of the ignorance of the interstellar magnetic field.
We point out that if in addition a deficit of the neutronization burst
neutrinos is established, it would be possible to extract the value of the
magnetic moment from observations. We also briefly discuss a possible Majorana
magnetic moment signature due to \nu_e -> \bar{\nu}_{\mu,\tau} flip inside the
supernova core.
