The neutronization burst phase of a core-collapse supernova, which lasts for a few tens of milliseconds post-bounce, is dominated by electron neutrinos and can offer exceptional discovery potential for neutrino transition magnetic moments. We simulate the neutrino spectra from the burst phase in forthcoming neutrino experiments like the Deep Underground Neutrino Experiment, and the Hyper-Kamiokande, by taking into account spin-flavour conversions of supernova neutrinos caused by interactions with ambient magnetic fields. We find that the sensitivities to neutrino transition magnetic moments are an order to several orders of magnitude better than the current terrestrial and astrophysical limits. We also discuss how this realization might provide light on the Dirac/Majorana nature and the neutrino mass-generation mechanism.