Sterile neutrinos with masses up to (100)  MeV can be copiously produced in a supernova (SN) core through the mixing with active neutrinos. In this regard, the SN 1987A detection of neutrino events has been used to put constraints on active-sterile neutrino mixing, exploiting the well-known SN cooling argument. We refine the calculation of this limit including neutral current interactions with nucleons, which constitute the dominant channel for sterile neutrino production. We also include, for the first time, the charged current interactions between sterile neutrinos and muons, relevant for the production of sterile neutrinos mixed with muon neutrinos in the SN core. Using the recent modified luminosity criterion, we extend the bounds to the case where sterile states are trapped in the stellar core. Additionally, we study the decays of heavy sterile neutrinos, affecting the SN explosion energy and possibly producing a gamma-ray signal. We also illustrate the complementarity of our new bounds with cosmological bounds and laboratory searches.