  We obtain generalizations of classical versions of the Weyl formula involving
Schr\"odinger operators $H_V=-\Delta_g+V(x)$ on compact boundaryless Riemannian
manifolds with critically singular potentials $V$. In particular, we extend the
classical results of Avakumovi\'{c} , Levitan and H\"ormander by obtaining
$O(\lambda^{n-1})$ bounds for the error term in the Weyl formula in the
universal case when we merely assume that $V$ belongs to the Kato class,
${\mathcal K}(M)$, which is the minimal assumption to ensure that $H_V$ is
essentially self-adjoint and bounded from below or has favorable heat kernel
bounds. In this case, we can also obtain extensions of the
Duistermaat-Guillemin theorem yielding $o(\lambda^{n-1})$ bounds for the error
term under generic conditions on the geodesic flow, and we can also extend
B\'erard's theorem yielding $O(\lambda^{n-1}/\log \lambda)$ error bounds under
the assumption that the principal curvatures are non-positive everywhere. We
can obtain further improvements for tori, which are essentially optimal, if we
strengthen the assumption on the potential to $V\in L^p(M)\cap {\mathcal K}(M)$
for appropriate exponents $p=p_n$.
