The best use case for MongoDB is as a document store. I can essentially cache numerous MySQL requests into a compiled set of useful information. Especially if the information changes somewhat infrequently, then instead of running MySQL requests for every page load I can pull the information from MongoDB. In most cases when I use MongoDB, its not as a persistent data store, but as a "compiled" data store.
MongoDB also has some useful set operations.
I for one don't believe that MongoDB is /directly/ competing with MySQL, Postgres, etc. but rather enhances these databases.