This was happening to me, too, on my iPhone 6 after the update to iOS 8.4. Start the Music app, see the splash screen, see a bit of network activity, then the network activity stops and the splash screen just shows forever. Kill the app and restart, same thing. Reboot the phone, same thing.
This only happened to me when I was not on WiFi, and it may have been the first time I tried to use the Music app after the upgrade. I also have the "Use Cellular Data" option disabled under Settings | iTunes & App Store.
I was able to get back to my music using the technique detained below, but for some users it may represent a hardship due to their own uses of the application, etc. Here's what I did:
1. Go to Settings | Music and switch "Show Apple Music" off.
2. Re-start the Music app. I got right into my music.
Re-enabling the "Apple Music" setting above caused the Music app to hang again on the splash screen.
While writing-up this reply, I decided to play-around a bit. I re-enabled "Apple Music" from Settings, got on a WiFi network, and launched the Music app. I immediately got the "Sign up for Apple Music" screen after the splash screen. So it appears that the combination of "[Don't] Use Cellular Data" for iTunes and not being on a WiFi network causes the Music app to hang. Possible remedies are:
a. Enable "Use Cellular Data" under Settings | iTunes & App Store
b. Disable "Show Apple Music" under Settings | Music
c. Get on a WiFi network and decline the 30-day trial of Apple Music
At this point, even disabling WiFi and having "Show Apple Music" enabled allows the Music app to start right up, and I can play music as usual.
Hope that helps.