  A dense neutrino gas, such as the one anticipated in the supernova
environment, can experience fast neutrino flavor conversions on scales much
shorter than those expected in vacuum probably provided that the angular
distributions of $\nu_e$ and $\bar\nu_e$ cross each other. We perform a
detailed investigation of the neutrino angular distributions obtained by
solving the Boltzmann equations for fixed matter profiles of some
representative snapshots during the post-bounce phase of core-collapse
supernovae in multidimensional calculations of an $11.2\mathrm{M}_{\odot}$ and
a $27\mathrm{M}_{\odot}$ progenitor models. Although the
$11.2\mathrm{M}_{\odot}$ model features $\nu_e - \bar\nu_e$ angular crossings
and the associated fast modes at different time snapshots, the
$27\mathrm{M}_{\odot}$ model does not show any crossings within the decoupling
region.
  We show that this can be understood by studying the multipole components of
the neutrino distributions.
  In fact, there is a higher chance for the occurrence of $\nu_e - \bar\nu_e$
angular crossings for the zones where the multipole components of the neutrino
distributions are strong enough. We also show that there can exist more than
one crossings between the angular distributions of $\nu_e$ and $\bar{\nu}_e$.
In addition, apart from the crossings within the neutrino decoupling region,
there is a class of $\nu_e-\bar\nu_e$ angular crossings which appears very deep
inside the proto-neutron star.
