  The large neutrino fluxes emitted with a distinct flavor hierarchy from
core-collapse supernovae (SNe) during the post-bounce accretion phase, offer
the best opportunity to detect effects from neutrino flavor oscillations. We
perform a dedicated study of the SN neutrino flavor evolution during the
accretion phase, using results from recent neutrino radiation hydrodynamics
simulations. In contrast to what expected in the presence of only
neutrino-neutrino interactions, we find that the multi-angle effects associated
with the dense ordinary matter suppress collective oscillations. This is
related to the high matter densities during the accretion phase in
core-collapse SNe of massive iron-core progenitors. The matter suppression
implies that neutrino oscillations will start outside the neutrino transport
region and therefore will have a negligible impact on the neutrino heating and
the explosion dynamics. Furthermore, the possible detection of the next
galactic SN neutrino signal from the accretion phase, based on the usual
Mikheyev- Smirnov-Wolfenstein effect in the SN mantle and Earth matter effects,
can reveal the neutrino mass hierarchy in the case that the mixing angle
$\theta_{13}$ is not very small.
