  We study the spin-flavor transitions of neutrinos in the magnetic fields of a
collapsing star. For the neutrino mass squared difference (10^{-10} - 10) eV^2
the transitions take place in almost isotopically neutral region of the star,
where the effective matter density is suppressed up to 3 - 4 orders of
magnitude. This suppression is shown to increase the sensitivity of the
neutrino bursts studies to the magnetic moment of neutrino by 1.5 - 2 orders of
magnitude, and for realistic magnetic field the observable effects may exist
for (2 - 3)10^{-14} Bohr magneton. In the isotopically neutral region the jumps
of the effective potential exist which influence the probabilities of
transitions. The experimental signatures of the spin-flavor transitions are
discussed. In particular, in the case of direct mass hierarchy, the spin-flip
effects result in a variety of modifications of the electron antineutrino
spectrum. Taking this into account, we estimated the upper bounds on the
magnetic moment from the SN1987A data. In the isotopically neutral region the
effects of possible twist of the magnetic field on the way of neutrinos can be
important, inducing distortion of the neutrino energy spectra and further
increasing the sensitivity. However, if the total rotation angle is restricted
the absolute change of probabilities is small.
