We obtain generalizations of classical versions of the Weyl formula involving Schrödinger operators on compact boundaryless Riemannian manifolds with critically singular potentials V. In particular, we extend the classical results of Avakumović [1956], Levitan [1952] and Hörmander [1968] by obtaining bounds for the error term in the Weyl formula in the universal case when we assume that with the negative part belongs to the Kato class, which is the minimal assumption to ensure that HV 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 [1975] theorem yielding bounds for the error term under generic conditions on the geodesic flow, and we can also extend Bérard’s (1977) theorem yielding 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 and for appropriate exponents p = pn .