Get the fingerprint sensor working on that fancy new iPhone you bought
Visiting family over Thanksgiving, I saw my sister was carrying a brand-spankin'-new iPhone 5s. She and her husband had changed carriers and got the new phones as part of the deal.
("iPhone 5s" still looks to me like the plural of "iPhone 5." What are the plurals of those supposed to be? "iPhone5ses" and "iPhone 5ives"? But I digress.)
We were talking about how she liked it, and I asked her about the fingerprint sensor on the home button. She said, a little sheepishly, that she hadn't set it up. It turns out the two of them had upgraded before changing carriers, and the iPhone 5s (the new one, singular) she'd been given by her old carrier didn't unlock when she touched the sensor. She'd almost always end up making multiple attempts and then being prompted to enter her passcode, so eventually she just turned TouchID off and used the passcode. When they changed carriers and they got their new phones, she'd just adopted the same setup.
Apparently, it's not an uncommon problem — and it's usually easily resolved, whether by adjusting the way you use it or exchanging your still-new phone for one that works. Wired has a pretty good article that explains the problem in some very technical terms before giving a pretty straightforward explanation of what you need to do differently:
First, for it to work properly, your finger needs to make contact not just with the sapphire of the home button, but also the stainless steel ring surrounding it. Next, the sensor itself works by measuring electrical differences between the ridges and valleys of your fingerprints. If your hands are too dry, it’s going to be difficult for your print to be recognized (this could be a growing problem in the dry winter months ahead). Conversely, if your fingers are too moist or oily, recognition can also fail, as those valleys get filled. If the button gets dirty, as it likely will over time, you’ll also want to clean it to keep Touch ID working properly. Apple suggests using a clean, lint-free cloth.
Those tips apply both to when you're attempting to unlock your phone and to when you're scanning your finger to set up Touch ID. For what it's worth, Apple also has its own guide for using TouchID that offers some of the same advice, and adds a couple hints for making sure more of your fingerprint is captured during setup.
But if all else fails and it seems clear the problem isn't you, you should definitely exchange that iPhone 5s (or 5sses, if you own more than one) for one that works correctly. Those suckers cost too much to just accept that a flagship feature isn't going to work the way it should.