I began experiencing this issue a couple weeks after buying my iPhone 5s and was totally puzzled about it. I almost read the whole discussion, and tried several of the suggested methods. In the end, what worked for me was changing the way I use Touch ID.
I'm not sure why the way I used it in the begining tends to work for a couple hours and then it stops to, but now it's working flawlessly for me. For what I read on the discussion, almost all of us have experienced the same problem, and my solution was quite simple: instead of using the top of my finger, I started using the center (where the fingerprint actually is). I sounds obvious, but the thing is that you have to put your finger in a different position than in previous iOs devices.
Maybe this discussion has become so long and complex because there are several different causes involved: in the begining it might have been software (for what I read, there was a glitch, but was corrected long ago), for some may be hardware, but for some others (like me) it may be just learning how to use it correctly.
I hope my experience help others overcome this annoying issue.
(sorry about any mistake I might have commited, but was you can obviously tell, I'm not a native english speaker)