Graph Has Two / Three Different Minimal Spanning Trees?
Kapoor &amp; Ramesh (proper SIAM J. Comput. version, free(?) personal website version) give an algorithm for enumerating all minimum spanning trees in both weighted and unweighted graphs.

My understanding of the basic idea is that you start with one MST, then swap edges that lie along cycles in the graph (so as long as the weights are okay, you're replacing one edge with another that you know will reconnect the tree).

For the weighted case they give a running time to list all minimum spanning trees of $O(N\cdot|V|)$ where $N$ is the number of such spanning trees. It enumerates them in order of increasing weight, and my current (cursory) understanding suggests that it's perfectly feasible to terminate the algorithm after it has generated a given number of trees (as it just starts with the MST and produces them sequentially).