Neutrino electromagnetic properties are of great importance from the viewpoint of fundamental theory, and applications as well. It is of common knowledge that neutrinos determine the dynamics of supernova explosion to a large extent. In this work we study the effect of matter polarized by an external magnetic field on neutrino spin evolution and propagation inside supernovae. Alternatively, the problem of neutrino interaction with such kind of matter can be treated as the interaction of the induced neutrino magnetic moment (IMM) with the magnetic field. Using the corresponding interaction Lagrangian we obtain the effective evolution equation for a neutrino with IMM and, on its basis, consider neutrino spin oscillations for different cases of Dirac/Majorana neutrino type, absence/presence of neutrino anomalous magnetic moment (AMM). It is shown that due to IMM the neutrino flux from a supernova undergoes additional attenuation. Also, the effects of IMM and AMM when taken together can cancel each other leading to a specific maximum in the neutrino spectrum from supernovae.