Based on the shell model for Gamow-Teller and the random phase approximation for forbidden transitions, we calculate cross sections for inelastic neutrino-nucleus scattering (INNS) under supernova (SN) conditions, assuming a matter composition given by nuclear statistical equilibrium. The cross sections are incorporated into state-of-the-art stellar core-collapse simulations with detailed energy-dependent neutrino transport. While no significant effect on the SN dynamics is observed, INNS increases the neutrino opacities noticeably and strongly reduces the high-energy tail of the neutrino spectrum emitted in the neutrino burst at shock breakout. Relatedly the expected event rates for the observation of such neutrinos by earthbound detectors are reduced by up to about 60%.
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