  Neutrino with magnetic moment can experience a chirality flip while
scattering off charged particles. This effect may lead to important
consequences for the dynamics and the neutrino signal of the core-collapse
supernova. It is known that if neutrino is a Dirac fermion, then nu_L->nu_R
transition induced by the chirality flip leads to the emission of sterile
right-handed neutrinos. The typical energies of these neutrinos are rather
high, E ~ (100-200)MeV. Neutrino spin precession in the magnetic field either
inside the collapsing star or in the interstellar space may lead to the
backward transition, nu_R->nu_L. Both possibilities are known to be
interesting. In the former case high-energy neutrinos can deliver additional
energy to the supernova envelope, which can help the supernova to explode. In
the latter case high-energy neutrinos may be detected simultaneously with the
"normal" supernova neutrino signal, which would be a smoking gun for the Dirac
neutrino magnetic moment. We report the results of the calculation of the
supernova right-handed neutrino luminosity up to 250 ms after bounce. They
allow to refine the estimates of the energy emitted in right-handed neutrinos.
Also the sensitivity of water Cherenkov detectors to the Dirac neutrino
magnetic moment is estimated. For mu_Dirac=10^{-13}mu_B Super-Kamiokande is
expected to detect at least few high-energy events from a galactic supernova
explosion.
  Also we briefly discuss the case of Majorana neutrino magnetic moment. It is
pointed out that spin flips may quickly equilibrate electron neutrinos with
non-electron antineutrinos if mu_Majorana~10^{-12}mu_B. This may lead to
various consequences for supernova physics.
