We analyze the possibility of probing nonstandard neutrino interactions (NSI, for short) through the detection of neutrinos produced in a future galactic supernova (SN). We consider the effect of NSI on the neutrino propagation through the SN envelope within a three-neutrino framework, paying special attention to the inclusion of NSI-induced resonant conversions, which may take place in the most deleptonized inner layers. We study the possibility of detecting NSI effects in a Megaton water Cherenkov detector, either through modulation effects in the e spectrum due to (i) the passage of shock waves through the SN envelope, (ii) the time dependence of the electron fraction, and (iii) the Earth matter effects; or, finally, through the possible detectability of the neutronization e burst. We find that the e spectrum can exhibit dramatic features due to the internal NSI-induced resonant conversion. This occurs for nonuniversal NSI strengths of a few %, and for very small flavor-changing NSI above a few 10 ÿ5 .