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With the complete redesign featuring the new streaming mixin (which I don't use), it seems that when listening to music from my library (stored on my device) I can only shuffle songs within a single album for an artist.
This is not a dupe of this question, in that I'm not asking how to shuffle all songs in the library.
Instead of selecting the artist to display the albums and songs of the artist, tap the image of the artist and the artists songs will play shuffled.
Answer: Siri > Play " artist " songs shuffle
These seem to be the current ways to shuffle in the new 8.4 Music.app: About.com "How to Shuffle Music on the iPhone" - Updated 7/8/2015
Shuffle All Songs
To shuffle all the songs in your Music app library, which will give you the greatest variety and a roughly random order of songs, follow these easy steps:
Open the Music app
Tap My Music
Make sure the menu just below Apple Music is set to Songs
Find a song you'd like to listen to and tap it to begin it playing
Tap the playback bar just above the buttons at the bottom of the screen to reveal the album art and full set of Music app controls
At the bottom of the screen, tap the Shuffle button, which is two arrows crossing over each other. You'll know it's enabled when it's highlighted.
With that done, your next song will be randomly chosen and you're off on a shuffling adventure. Use the forward arrow to skip to the next song or the back arrow to return to the last one.
To turn off shuffle, just tap the two-arrow icon again so it is no longer highlighted.
This is an indirect method, absent an actual shuffle button on the artist's page, that works for me:
You'll now be listening to a shuffle of all songs by that artist. Definitely less convenient, but it seems to work.
To confirm that you added all the songs you want, click the now playing banner and click the "menu" icon to the right of the "next song" arrows, this will bring up all the songs that are up next.
If that's too lengthy, you can use Siri to play songs by an artist - one tap to accomplish.
Siri play songs by U2 shuffle
You can also make a playlist with the artists you desire and play those by tapping or by voice control.
NOT IMPOSSIBLE to shuffle all your music unless you have an Internet connection. (i.e. It IS possible to shuffle all songs, though it's not exactly intuitive.)
Procedure:
1) Open music app.
2) Open My Music (left button at bottom of screen).
3) Press Songs in middle of screen (in list of Artists, Albums, etc.) You may need to scroll to top of page.
4) Press album cover icon at far left of any song in the list.
5) Press "now playing" bar near bottom of screen (directly above My Music, Playlists, Radio, etc.) at very bottom of screen. Album cover and full transport controls will appear.
6) Press shuffle button at bottom left/center of screen (below horizontal volume bar).
7) Press Next transport button (on right side of Play button). All songs in phone will play in random (shuffled) order.
If you are willing to use alternative players I found this app on the App Store called Ecoute. It seamlessly works with your iTunes music library and you can shuffle on artist, genre and more. It is a basic, easy and intuitive player...
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FOund this in another forum, by another person and it works:
Go into your music, type the artists name in the search bar and make sure that 'my music' is selected. then the artist will come up, their albums will come up, then their songs will come up. Click on songs and once you play one you can press shuffle and it will shuffle all of the songs from that artist.
As with many of the other responses, I have found the easiest method for shuffling all songs for an artist with multiple albums:
-My Music -Make sure Artists is selected -Scroll to your artist -Tap the icon to the left of the name of the artist - this will usually be a thumbnail graphic of an album cover -Tap the song bar towards the bottom of the screen -Tap the "shuffle" icon at the bottom - this will look like two crossing arrows
It's impossible to shuffle all your music unless you have an internet connection, which is the most ridiculous thing in a long line of bad design moves from Apple.
I've moved across to this free app https://itunes.apple.com/au/app/musis-a-simple-music-player/id849003298?mt=8 which works very nicely with your existing music library, has very intuitive controls, shuffles, and no it's not in Korean despite the screenshots!
tell Siri to play my music on shuffle.
