  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. Our main motivation
comes from the construction of singular Riemannian metrics with prescribed
non-classical Weyl's law. Namely, for any non-decreasing slowly varying
function $\upsilon$ we construct a singular Riemannian structure whose spectrum
is discrete and satisfies \[ N(\lambda) \sim \frac{\omega_n}{(2\pi)^n}
\lambda^{n/2} \upsilon(\lambda). \] Examples of slowly varying functions are
$\log\lambda$, its iterations $\log_k \lambda = \log_{k-1}\log\lambda$, any
rational function with positive coefficients of $\log_k \lambda$, and functions
with non-lo\-ga\-rithmic growth such as $\exp\left((\log \lambda)^{\alpha_1}
\dots (\log_k \lambda)^{\alpha_k} \right)$ for $\alpha_i \in (0,1)$. A key tool
in our arguments is a new quantitative estimate for the remainder of the heat
trace and the Weyl's function on Riemannian manifolds, which is of independent
interest.
