In this paper, we study the asymptotic growth of the eigenvalues of the
Laplace-Beltrami operator on singular Riemannian manifolds, where all
geometrical invariants appearing in classical spectral asymptotics are
unbounded, and the total volume can be infinite. Under suitable assumptions on
the curvature blow-up, we show how the singularity influences the Weyl's
asymptotics and the localization of the eigenfunctions for large frequencies.
As a consequence of our results, we identify a class of singular structures
such that the corresponding Laplace-Beltrami operator has the following
non-classical Weyl's law: \[ N(\lambda) \sim \frac{\omega_n}{(2\pi)^n}
\lambda^{n/2} \upsilon(\lambda), \] where $\upsilon$ is slowly varying at
infinity in the sense of Karamata. Finally, for any non-decreasing slowly
varying function $\upsilon$, we construct singular Riemannian structures
admitting the above Weyl's law.
A key tool in our arguments is a universal estimate for the remainder of the
heat trace on Riemannian manifolds, which is of independent interest.