Is there a general (and efficient) method to determine how many graphs there are with a given degree sequence $S$? How many isomorphism classes are there among those?
Is the question different for digraphs, i.e. how many directed graphs are there with a degree sequence $S$ (list of pairs of integers)?
Tentative answer: As an approximation, one could count the number of (non-isomorphic) graphs of $n$ nodes divided by the number of distinct degree sequences for graphs of $n$ nodes.
Another approach is to generate many random graphs with degree sequence $S$ (e.g. with Havel-Hakimi), and determine the fraction of isomorphic classes. However this is a very rough estimation and would be inefficient for large $n$. Moreover, I am not sure that Havel-Hakimi or similar techniques cover all possible graphs, and whether they do so with uniform probability.