The expression language operates only on individual files, so no expression can tell you directly things like how many artists, albums or tracks exist. You can get the information indirectly by creating a list with one of each value of interest, like one track from each artist (i.e., ~nodup=[Artist]). The answer with then be the number of files in the list—which appears in the status bar at the bottom.
The easier solution is just to use a Panes View. The first item in each pane shows the number of unique values in that category for the current collection. That will work for [Artist], but not for [Album] if you have multiple albums named things like "Greatest Hits." That's easily solved by using instead a pane for [Artist]\[Album].
The screenshot illustrates both solutions in one view. I created Smartlists for unique values of Artist, Album and Track. These automatically appear in my Playlists pane. I've made pane selection of Rock Artists with Single-artist albums, and see I have 193 such artists.