hey folks....
16GB black iPhone 4S...Canadian provider Telus...serial #C39xxx series.
I, too, could literally watch the battery drop 1% every 5-ish minutes when I first got the phone. Even placing phone in Airplane mode caused the same swift drop of power. .. However, this happened -after- restoring from a backup of an iPhone 4.
I tried out the solution(s) provided here...which was to factory restore and turn off all System Services...iCloud (doesn't matter to me since everything I need is saved in iTunes anyway)...and charge the phone to 100% and then some using Battery Doctor.
After doing this, in Airplane mode..with approx. 40 minutes of basic usage (just moving around through menus, playing an audio book in iTunes..no other apps running..no surfing..still in airplane mode)...and just over 10 hours standby time...90% charge left...got to class, turned Airplane mode off..and could see the battery drop more quickly..but, then..around Noon-1pm..something happened..which was the battery reached around 65%..and has finally just now dropped to 64% 5 hours later. This was with using Siri 2-3x..sent about 4 texts..snapped several photos...and made a 5 minute phone call, checked Voice Mail a couple of times..and more iTunes use.
So, hmmm...maybe for some of us (at least me) the problem was just letting the battery stabilize (which doesn't make scientific sense to me) with time. Certainly, though, restoring from an old iPhone 4 backup f**'d my battery usage over! Won't be doing that again.
Good luck, everyone.
edit: it seems the (grudging?) consensus in the very long Apple support thread re: the battery issue is that one must drain battery, charge, drain, charge 2-3x for the "heat" issues and the short battery life problems to disappear.