How can I rank paths through an HMM?
Use log-probabilities, then use k-shortest paths.

The probability of a path is the product of the probabilities on its edges.  If you log-transform everything (so that each edge is annotated with the log of the probability rather than the probability itself), the log-probability of a path becomes the sum of the log-probabilities on the edges.

Now you want to rank the paths by decreasing log-probability, and find the $k$ paths with the highest log-probability.  Note that the log-probability of a path is just the length of the path (the sum of the lengths of the edges).

Therefore, the problem becomes: given a graph with a length on each edge, find the $k$ shortest paths.

There are efficient algorithms for finding the $k$ shortest paths.  See, e.g., https://cstheory.stackexchange.com/a/32501/5038.