I had a 14 month old iPhone 5, and noticed recently (last month or so) that the battery was draining rapidly. It would shut off sometimes at 30% and would lose large chunks at a time as others have pointed out. When I upgraded to iOS7, my battery life seemed to actually improve (this was prior to the issues above that I have been having). However, once I went to 7.0.3, I really started to notice poor battery life.
I don't have AppleCare, so I decided to make an appointment at the Genius Bar at the Apple Store near me. While I was in the store waiting, I sent my wife an email with a photo attachment that was about 2 MB, and the battery dropped over 20%. I knew something was wrong (I have just about everything turned off that could cause excessive battery drain).
They ran a diagnostic on my battery and said that it was at the end of its lifecycle - i.e. in really bad shape, and needed to be replaced. They told me they could put a new battery in my phone for $79. Said it would take about 20-30 minutes. I left, then came back and it still wasn't ready...and I waited, and waited and waited...FINALLY, they came out with a white box, along with my phone, and said they tried to put a new battery in my existing phone, and it didn't work properly. Therefore, they would give me a replacement phone for $79 (apparently normally it costs $239). I know there has been some question about whether these phones are refurbished or not...I have no idea, BUT this new phone has been holding the charge WAY better than my old phone. I'm running 7.0.4. I generally can go all day without having to charge (about 6 - 7 hours of usage and 18+ hours of standby). Plus, now I can run the battery down to 1% and it still won't shut off by itself. The guy at the store suggested I actually turn off my phone two or three times a week - said it helps overall with my phone.
My point is this - if you're having the types of issues that are detailed in this thread, you should at least take your phone in and have them run diagnostics on it. You could be the same situation that I"m in - I plan on getting the new iPhone next year and this replacement iPhone 5 will hold me over till then, hopefully with no issues. That was $79 well spent, in my opinion...well worth it to have my normal battery life back.
Good luck!