We study CP-violation effects when neutrinos are present in dense matter, such as outside the proto-neutron star formed in a core-collapse supernova. Using general arguments based on the Standard Model, we confirm that there are no CP-violating effects at the tree level on the electron neutrino and anti-neutrino fluxes in a core-collapse supernova. On the other hand significant effects can be obtained for muon and tau neutrinos even at the tree level. We show that CP-violating effects can be present in the supernova electron (anti-)neutrino fluxes as well, if muon and tau neutrinos have different fluxes at the neutrinosphere. Such differences could arise due to physics beyond the Standard Model, such as the presence of flavor-changing interactions.