A couple important things to note: There are two main issues people on this thread are having. One involves how the user is operating the device, i.e. user error. You must set up Touch ID properly in the scanning phase, and then present your print properly for it to work. There is good advice in this thread in that regard--like making sure you contact the metal ring, fingers not too wet or dry--that ought to teach you how to operate Touch ID properly. Once you've done that, it should work almost every time.
The second issue--whether others wants to acknowledge it or not--is those people for which Touch ID works fine for a while and then suddenly stops working. That's the problem I had, and I see from the descriptions here that several others experienced the same issue. If you solved your issue just by learning how to operate Touch ID properly, that's great. You can ignore what follows. For those that are using it properly, where it will work fine for a while and then suddenly fail, this is for you:
I've done a lot of troubleshooting with this, both on my own and with the Apple Genius at my local store. It's most certainly a software issue. What we found was that certain apps, both native iOS 7 apps and third party apps (particularly third party apps) cause a lot of crashing. Check your diagnostic log (Go to settings-> general-> about-> diagnostics and usage-> diagnostic and usage data). Apps crashing, the phone randomly resetting, etc. can cause Touch ID to temporarily stop working. If you see a lot of app crashes, backboard crashes and low memory warnings, then these steps might work for you. You can try these in order, because you may not have to do all of them.
1. Reset the phone by pressing home and sleep button together until you see Apple logo.
2. Close any apps that have been crashing (double-click home button and swipe the window for that app up) then reset phone by pressing home and sleep button together until you see Apple logo.
3. If you have multiple windows open in Safari close all of them. This causeed a low memory warning for me.
4. Delete any third party apps that are crashing regularly and re-download them, reset phone.
5. Turn off background app refresh (settings-> general-> Background App Refresh) or you can selectively do it for problem apps, reset phone.
6. Diable any crashing apps in notification center and/or location services, reset phone.
7. Last resort - plug phone into iTunes, do a backup, restore software and set up as a new phone. Don't reinstall all the apps, partciulcalry those that have been crashing. Run it for a while and see how it works, then add troublesome apps back one-at-a-time.
iOS 7 is really buggy, particularly with third party apps. That diagnostic log is invaluable for troubleshooting and should help you pinpoint problems. I had Touch ID working perfectly for four days, the OS crashed while I was reviewing photos in the native photos app (meaning all the phone reset itself and the Apple logo came up). Suddenly, Touch ID wasn't working again. Looked at the log and saw a bunch of crashes and low memory warnings. I went in and closed all open apps, cleared out open Safari windows, resset the phone. Touch ID magically worked again. You should not have to re-register your prints.
This, of course, is a bit of a bandaid because the next crash will cause it to go haywire again. Apple software engineers (and the app developers) need to work on iOS 7 to get this right. Hopefully an update is coming soon that will fix this and many other bugs.