Neutrino self-interactions are known to lead to non-linear collective flavor oscillations in a core-collapse supernova. In this talk, I will point out new possible effects of non-standard self-interactions (NSSI) of neutrinos on flavor conversions in a two-flavor framework. I will show that, for a single-energy neutrino-antineutrino ensemble, a flavor instability is generated even in normal hierarchy for large enough NSSI. Using a toy model for the neutrino spectra, I will demonstrate that flavor-preserving NSSI lead to pinching of spectral swaps, while flavor-violating NSSI cause swaps to develop away from a spectral crossing or even in the absence of a spectral crossing. Consequently, NSSI could give rise to collective oscillations and spectral splits even during neutronization burst, for both hierarchies.