Counting/Enumerating Minimal Edge Covers
Counting edge covers is #P-complete. You can find a proof here Graph Orientations with No Sink and an Approximation for a Hard Case of #SAT. See also the stack-exchange discussion. One can probably modify the proof to show #P-completeness for minimal edge covers.
enumerating (minimal) edge covers can be done with polynomial delay as the hypergraph has bounded degree (see New Results on Monotone Dualization and Generating Hypergraph Transversals)
the following Low-Exponential Algorithm for Counting the Number of Edge Covers on Simple Graphs claims an $O(1.465575^{(m−n)} × (m + n))$ time algorithm. I am wondering whether one cannot do better.