One possible cause is that iTunes Match is not enabled even though you think it is. This happened to me on my iPad. I launched Settings and selected to turn on iTunes Match (you can do so either from the settings for iTunes and App Stores or for Music). While this appears to work, if you exit and return to the setting, you'll find that the iTunes Match option has reverted to Off. If your Music library is entirely based on iTunes Match, this would account for why nothing is available.
The simplest fix, if it works, is to go to Settings -> General -> Reset and select Reset Network Settings. This did the trick for me. I had to rejoin my Wi-Fi network. After that, when I next enabled iTunes Match, it stayed enabled. The Music app now showed all the music that was in my iCloud storage.
If this fix doesn't do the job, you might have success with more extreme solutions. One such recommended fix involves connecting your iPhone to your Mac and, using a Mac utility such as PhoneView, navigating to Disk -> iTunes_Control -> iTunes. From here, delete the three MediaLibrary files. Before you try this, check the postings in this thread or the one linked above for more details.
Changes to iTunes Match
Speaking of iTunes Match, when you first enable the feature on your iOS device, you will likely see an alert that states: "iTunes Match will replace the music library on this device." This means that any music previously synced to your iOS device via iTunes on your Mac will be removed. Doing so also disables Genius Mixes and Genius Playlists.
This is different than how iTunes Match worked prior to iOS 6. Originally, you could enable iTunes Match and still retain music locally stored on your iOS device. No more. Now it is one or the other. Some cases of "missing music" may relate to previously synced music being removed.
This is not the only unwelcome change in iTunes Match. As noted in Macworld's look at iOS 6's Music app, you can no longer download individual songs from iCloud; it's either an entire album, playlist, artist's catalog, or nothing.
"Incompatible" third party apps
As with any iOS upgrade, numerous third-party apps will require an update to be compatible with the new OS. If, after updating to iOS 6, an app starts crashing on launch or exhibiting other strange symptoms, make sure you have the latest version of the app installed.
If you do have the latest version, and the problem persists, delete the app from your iOS device (making sure you first have any associated documents and data backed up) and then re-download the app. This fixed a problem I was having with errors in Navigon.
If neither of these tips work, report the problem to the app's vendor and check if they have a bug-fix update coming soon.
Problems with Apple's iOS 6 apps
Problems have been reported with several of the new or updated apps included as part of iOS 6. These include Passbook not working as expected (as I've covered here) and the Podcasts app not showing podcast playlists (as noted in Mac OS X Hints). Oh, and you may have heard about concerns with Maps.