The study of neutrino flavor evolution in supernovae has recently pointed out that relaxing the symmetries of the standard bulb model, which was commonly adopted until a few years ago for describing neutrino emission and propagation, might introduce instabilities in flavor space, which are exponentially growing in both time and space. In this talk I focus on two kind of instabilities. First I show that space and time inhomogeneities can excite flavor conversions at small spatial scales and can spontaneously generate a pulsating component in the flavor composition. In particular, the pulsation can compensate the suppression of flavor conversions induced by large matter effects. Then, I will discuss the so called “fast flavour conversions”, which develop a few meters from the supernova core, introducing both decoherence and flavour equilibration. If confirmed, both instabilities may have an important role in supernova explosion, such as in the shock reheating and in nucleosynthesis processes.