Counting Minimum Spanning Trees
The best exposition on how to count the number of minimum spanning trees is, as far as I have seen, a stackoverflow answer by j_random_hacker.

In the course to answer a different question, that answer explains very well an algorithm that counts the number of MSTs.


It establishes that Kruskal's algorithm can find every MST. 
It breaks up the Kruskal's algorithm into a series of blocks, each of which consists of a sequence of adding the edges of the same weight into a component multigraph that has been built in the previous block of operation.
It proves that the number of MSTs is the product of the number of spanning forests in the multigraph for each block-defining weight. 
Finally, the number of spanning forests for a multigraph can be computed by Kirchhoff's theorem.