Like many things in, the shuffle button is buried amongst the clutter. When you are viewing a playlist or an album, the shuffle button is right where you would expect it -- easily accessible at the top of the of the list of tracks. If you want to shuffle all the songs from your music library or, say, a genre or an artist, however, the shuffle button is dastardly hidden.
First up, shuffling all songs.
Tap the My Music tab. From the Library view (as opposed to Playlists), choose Songs from the header below the Recently Added belt. You will now have a list of songs at your fingertips. Tap a song to begin playing it and then tap the mini-player to expand it. From the expanded mini-player, you can tap the shuffle button at the bottom, letting you listen to a shuffled mix of your songs rather than alphabetically by song title.
Next up, starting a genre shuffle. Or a shuffle by artist.
The tricky thing about starting a genre shuffle is actually starting to play a genre. From the Library view of the My Music tab, choose Genre (or Artist) from the header below the Recently Added belt. If you tap on the name of a genre from the list, you don't begin playing the whole genre. Instead, you are taken to a list of the albums from your library in that genre, which lets you play a single album but not all of them. The trick is to tap the thumbnail for a genre instead of its name from the list of genres. Tapping on the thumbnail begins playing all of the songs of a genre. Then the shuffle process is the same as shuffling all songs -- once you have started playing a genre, tap the mini-player to expand it and then tap the shuffle button.
Alternatively, you can employ Siri to shuffle whatever it is you are currently listening to in Apple Music instead of hunting for the shuffle button. When you are listening to a selection of songs that you'd like shuffled, you can simple tell Siri "shuffle" and she'll hop to it.
In related Apple Music news, learn how to replay Beats 1 radio shows you missed, juggle tracks with Up Next and set your iPhone alarm to wake you to a song from Apple Music.