Number of Vertex Covers: when it is polynomial and when it is superpolynomial
There's a close relation between the independence number of a graph and the number of its vertex covers, not close enough to completely answer your question but close: if the independence number is superlogarithmic then the number of covers is superpolynomial, and otherwise it's quasipolynomial.

More specifically if the independence number is $\alpha$, and the maximum independent set $I$ is the complement of a vertex cover $C$, then each of the $2^\alpha$ supersets of $C$ is also a vertex cover. So in this case if $\alpha \ge c\,\log_2 n$ then the number of covers is lower bounded by $n^c$. But each cover is complementary to an independent set and there can be only $O(n^\alpha)$ sets of at most $\alpha$ vertices, so if $\alpha \le c\,\log_2 n$ then this upper bound is $O(2^{c\, \log^2 n})$.