  Known classification results allow us to find the number of (equivalence
classes of) fine gradings on matrix algebras and on classical simple Lie
algebras over an algebraically closed field $\mathbb{F}$ (assuming
$\mathrm{char} \mathbb{F}\ne 2$ in the Lie case). The computation is easy for
matrix algebras and especially for simple Lie algebras of type $B_r$ (the
answer is just $r+1$), but involves counting orbits of certain finite groups in
the case of Series $A$, $C$ and $D$. For $X\in\{A,C,D\}$, we determine the
exact number of fine gradings, $N_X(r)$, on the simple Lie algebras of type
$X_r$ with $r\le 100$ as well as the asymptotic behaviour of the average, $\hat
N_X(r)$, for large $r$. In particular, we prove that there exist positive
constants $b$ and $c$ such that $\exp(br^{2/3})\le\hat
N_X(r)\le\exp(cr^{2/3})$. The analogous average for matrix algebras
$M_n(\mathbb{F})$ is proved to be $a\ln n+O(1)$ where $a$ is an explicit
constant depending on $\mathrm{char} \mathbb{F}$.
