  We present a review of a broad selection of nuclear matter equations of state
(EOSs) applicable in core-collapse supernova studies. The large variety of
nuclear matter properties, such as the symmetry energy, which are covered by
these EOSs leads to distinct outcomes in supernova simulations. Many of the
currently used EOS models can be ruled out by nuclear experiments, nuclear
many-body calculations, and observations of neutron stars. In particular the
two classical supernova EOS describe neutron matter poorly. Nevertheless, we
explore their impact in supernova simulations since they are commonly used in
astrophysics. They serve as extremely soft and stiff representative nuclear
models. The corresponding supernova simulations represent two extreme cases,
e.g., with respect to the protoneutron star (PNS) compactness and shock
evolution. Moreover, in multi-dimensional supernova simulations EOS differences
have a strong effect on the explosion dynamics. Because of the extreme
behaviors of the classical supernova EOSs we also include DD2, a relativistic
mean field EOS with density-dependent couplings, which is in satisfactory
agreement with many current nuclear and observational constraints. This is the
first time that DD2 is applied to supernova simulations and compared with the
classical supernova EOS. We find that the overall behaviour of the latter EOS
in supernova simulations lies in between the two extreme classical EOSs. As
pointed out in previous studies, we confirm the impact of the symmetry energy
on the electron fraction. Furthermore, we find that the symmetry energy becomes
less important during the post bounce evolution, where conversely the symmetric
part of the EOS becomes increasingly dominating, which is related to the high
temperatures obtained. Moreover, we study the possible impact of quark matter
at high densities and light nuclear clusters at low and intermediate dens
