  We establish some $C^{0,\alpha}$ and $C^{1,\alpha}$ regularity estimates for
a class of weighted parabolic problems in divergence form. The main novelty is
that the weights may vanish or explode on a characteristic hyperplane $\Sigma$
as a power $a > -1$ of the distance to $\Sigma$. The estimates we obtain are
sharp with respect to the assumptions on coefficients and data. Our methods
rely on a regularization of the equation and some uniform regularity estimates
combined with a Liouville theorem and an approximation argument. As a corollary
of our main result, we obtain similar $C^{1,\alpha}$ estimates when the
degeneracy/singularity of the weight occurs on a regular hypersurface of
cylindrical type.
