  We study the parabolic obstacle problem $$\lap u-u_t=f\chi_{\{u>0\}}, \quad
u\geq 0,\quad f\in L^p \quad \mbox{with}\quad f(0)=1$$ and obtain two
monotonicity formulae, one that applies for general free boundary points and
one for singular free boundary points. These are used to prove a second order
Taylor expansion at singular points (under a pointwise Dini condition), with an
estimate of the error (under a pointwise double Dini condition). Moreover,
under the assumption that $f$ is Dini continuous, we prove that the set of
regular points is locally a (parabolic) $C^1$-surface and that the set of
singular points is locally contained in a union of (parabolic) $C^1$ manifolds.

Analysis of PDEs