Beyond the stuff that has been mentioned the serious bug I've found is relating to music.
-Unable to put new songs from the desktop to the iPhone. iTunes lets me drag the songs onto the iphone, but instead of uploading the songs will just sit there doing nothing and the song will appear greyed out on the list in iTunes like it is still being uploaded - but it isn't.
Meanwhile, on the iPhone itself if you download using the iTunes store app it will download the song fine, but the cover art in the search menu's will be of a different song, and change every time you refresh the page.
Basically, there are some serious issues with the music app and with iTunes currently.
Mind you, this could be because I did an 'upgrade' instead of a restore. Because of this maybe there is a conflict with the old music playlists on there.
Other then that, works pretty good on my iPhone 5.
My banking app has some mild display issues, but it works good enough where it isn't a problem.
All the games I have work just fine as well. I suspect most games will work with iOS 7 without many problems since most draw their own menu's and what not.