  We introduce and develop the theory of Newton nondegenerate local Weil
divisors $(X,0)$ in toric affine varieties. We characterize in terms of the
toric combinatorics of the Newton diagram different properties of such singular
germs: normality, Gorenstein property, or being an Cartier divisor in the
ambient space. We discuss certain properties of their (canonical) resolution
$\tilde{X}\to X $ and the corresponding canonical divisor. We provide
combinatorial formulae for the delta--invariant $\delta(X,0)$ and for the
cohomology groups $H^i(\tilde X,\mathcal{O}_{\tilde X})$ for $i>0$. In the case
$\dim(X,0)=2$, we provide the (canonical) resolution graph from the Newton
diagram and we also prove that if such a Weil divisor is normal and Gorenstein,
and the link is a rational homology sphere, then the geometric genus is given
by the minimal path cohomology, a topological invariant of the link.
