Apple Music play count stats
Yes, you can see these details with a third party app


Genre
Rating
Track #
Composer
Lyrics
Date Added
Last Played
Last skipped
Play Count
Skip Count




SmartPlaylist on the App Store

SmartPlaylist Website

As of today the app costs $2.99 (in the US App Store). I bought it for the Smart Playlist functionality that I grew accustomed to with my iPod mini/iPod classic.

To see the play count and other statistics follow these steps


Play your desired song in the Music.app
Open SmartPlaylists.app
Tap Menu in the lower left
Select Now Playing
Double-tapping a field like "Skipped" or "Released" will show you the time as well.