  Recent theoretical work indicates that the neutrino radiation in
core-collapse supernovae may be susceptible to flavor instabilities that set in
far behind the shock, grow extremely rapidly, and have the potential to
profoundly affect supernova dynamics and composition. Here we analyze the
nonlinear collective oscillations that are prefigured by these instabilities.
We demonstrate that a zero-crossing in $n_{\nu_e} - n_{\bar{\nu}_e}$ as a
function of propagation angle is not sufficient to generate instability. Our
analysis accounts for this fact and allows us to formulate complementary
criteria. Using Fornax simulation data, we show that fast collective
oscillations qualitatively depend on how forward-peaked the neutrino angular
distributions are.
