  We present a comprehensive study on the occurrences of the collisional flavor
instability (CFI) and the fast flavor instability (FFI) of neutrinos based on a
two-dimensional (2D) core-collapse supernova (CCSN) simulation performed with a
Boltzmann radiation hydrodynamics code. We find that CFI occurs in a region
with the baryon-mass density of $10^{10}\lesssim \rho \lesssim
10^{12}\,\mathrm{g}\,\mathrm{cm}^{-3}$, which is similar to the previous
results in one-dimensional (1D) CCSN models. In contrast to 1D, however, the
CFI region varies with time vigorously in the 2D model, whereas it had a
quiescent structure in 1D. This is attributed to the fact that the turbulent
flows advected from a gain region account for the temporal variations. Another
noticeable difference from the 1D models is the appearance of resonance-like
CFI where number densities of $\nu_e$, $\bar\nu_e$ nearly coincide each other.
The CFI growth rate there is enhanced and can reach
$\sim10^{-3}\,\mathrm{cm}^{-1}$. As for FFI, on the other hand, it appears in
three different regions; (1) the region overlapped with the resonance-like CFI,
(2) neutrino decoupling regions where $\bar{\nu}_e$'s are strongly emitted, and
(3) optically thin regions where neutral-current scatterings dominate over
charged-current reactions. Although overall properties for FFI are consistent
with previous studies, we find that the number of electron-neutrinos lepton
number crossing (ELN crossing) temporary becomes multiple, which can be
assessed accurately only by multi-angle treatments in neutrino transport. We
find that the growth rate of FFI is always higher than CFI if both of them
occur, which suggests that the former is dominant for the linear evolution.
