Several users have reported that iTunes Mach is not working, meaning iTunes Match is not loading any music after updating to iOS 6. Users have also reported that when they launch iTunes Match, it says loading, but nothing happens (blank main screen and a picture of a cloud).
An Apple forum user suggested this, which seems to have fixed this issue:
1. Download and install iExplorer http://www.macroplant.com/iexplorer/
2. Plug in device first, then open iExplorer
3. Delete the iPhone\Media\iTunes_Control folder (you may want to copy this, just in case)
4. Restart the iPhone / iPad
5. Turn on iTunes Match in settings.
6. Start Music app