  A core-collapse supernova (SN) offers an excellent astrophysical laboratory
to test non-zero neutrino magnetic moments. In particular, the neutronization
burst phase, which lasts for few tens of milliseconds post-bounce, is dominated
by electron neutrinos and can offer exceptional discovery potential for
transition magnetic moments. We simulate the neutrino spectra from the burst
phase in forthcoming neutrino experiments like the Deep Underground Neutrino
Experiment (DUNE), and the Hyper-Kamiokande (HK), by taking into account
spin-flavour conversions of SN neutrinos, caused by interactions with ambient
magnetic fields. We find that the neutrino transition magnetic moments which
can be explored by these experiments for a galactic SN are an order to several
orders of magnitude better than the current terrestrial and astrophysical
limits. Additionally, we also discuss how this realization might shed light on
three important neutrino properties: (a) the Dirac/Majorana nature, (b) the
neutrino mass ordering, and (c) the neutrino mass-generation mechanism.
