  We consider the Laplace--Beltrami operator on a three-dimensional Riemannian
manifold perturbed by a potential from the Kato class and study whether various
forms of Weyl's law remain valid under this perturbation. We show that a
pointwise Weyl law holds, modified by an additional term, for any Kato class
potential with the standard sharp remainder term. The additional term is always
of lower order than the leading term, but it may or may not be of lower order
than the sharp remainder term. In particular, we provide examples of singular
potentials for which this additional term violates the sharp pointwise Weyl law
of the standard Laplace-Beltrami operator. For the proof we extend the method
of Avakumovi\'c to the case of Schr\"odinger operators with singular
potentials.
