  The goal of this paper is to explain how basic properties of perverse sheaves
sometimes translate via Riemann-Hilbert correspondences (in both characteristic
$0$ and characteristic $p$) to highly non-trivial properties of singularities,
especially their local cohomology. Along the way, we develop a theory of
perverse $\mathbf{F}_p$-sheaves on varieties in characteristic $p$, expanding
on previous work by various authors, and including a strong version of the
Artin vanishing theorem.
