  We consider the Schr\"odinger operators $H_V=-\Delta_g+V$ with singular
potentials $V$ on general $n$-dimensional Riemannian manifolds and study
whether various forms of pointwise Weyl law remain valid under this
pertubation. We prove that the pointwise Weyl law holds for potentials in the
Kato class, which is the minimal assumption to ensure that $H_V$ is essentially
self-adjoint and bounded from below or has favorable heat kernel bounds.
Moreover, we show that the pointwise Weyl law with the standard sharp error
term $O(\lambda^{n-1})$ holds for potentials in $L^n(M)$.
