How can I check how many of my mentions are of a certain type?

By default, the Mentions view shows you all types of mentions combined - posts, shares, comments, articles and messages sent to your authorized profiles. The number of current mentions is always shown after the 'Browse mentions' expression

If you are only interested in a certain type of mention, you must deselect all the irrelevant ones in the filtering section on the right side of the screen. The number of mentions will decrease and the used filter will appear above the sorting section.