Fingerprint registration guide: (updated 11/27/2013)
Please note that this guide should only be used if you experience Touch ID (TID) failing to work reliably hours or days after initially setting it up. Updates to this guide will be made as required.
Many Touch ID problems can be resolved by changing the way fingerprints are initially Enrolled.
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General info can be found in the Apple iPhone User Guide (PDF).
Other threads you may find helpful:
Make sure you're using the latest IOS, currently 7.0.4 as of 11/27/2013
Only Enroll (register) a given finger once. Make sure to delete it first if already registered. There's no need to record multiples of the same finger.
Don't "roll" the finger when registering as if making ink fingerprints on paper; It's not the same thing.
Keep your thumb or finger as flat as possible. Don't bend at the first joint. Make sure you're covering the outer home button ring fully.
Start the first register scan in the centerline of the finger, almost at the base of the nail, slightly towards the tip of the finger.
I make a total of 10 scans in 3 rows, 3 columns, and finally one last scan dead center of the 9.
Make three scans: finger center, left of center and right of center before starting to scan forward towards the fingertip.
With each scan, move slightly left and right, but not so far that the centerline of the finger moves off of the sensor. The idea is to keep a known, already scanned, "landmark" on the sensor with each scan during registration.
After the three scans in the first "row" are complete, move back to the centerline and slightly forward towards the tip to scan the second row, again moving only slightly, keeping the top portion of the last centerline scan on the sensor.
When prompted to "Adjust Your Grip" do not follow the on-screen display of the example finger position. You will find over time that it will fail. At this point, you can hit "back". You do not have to scan additional areas or wait for the "complete" message.
In no case should you bother with the very tip of the finger as I believe it's unreliable for those of us who have problematic fingers.
Don't exert a lot of pressure during the scan. Not too hard, not too soft.
Completely lift the finger straight up after the each training scan, move to the next spot, and place it straight down without sliding around or moving once you make contact. Lift your finger only after the phone vibrates.
This should give you a good starting point.
Using TID to unlock:
Keep the finger as flat as it was during registration using the center of the registered area. Briefly press the home button to wake the phone up, and ease up pressure while keeping the finger in full contact with the ring. If it fails to unlock, which it may if there was movement during the release, lift your finger and place it back down, making sure you are in the area you registered.
This guide does not address IOS components that are responsible for handling screen and home button operations from crashing or going brain-dead, most notably, but not limited to, having Twitter as the active app when the scree lock activates. This can cause TID to fail to read a registered finger, It will appear as if it's not even sensing that a finger is on the home button! This should be an easy S/W fix however, but is still "broken" in 7.0.4.
One workaround when this happens is to pull the Control Panel up from the lock screen, start then close the calculator. This action "resets" the focus back to the fingerprint sensor, and it should now work. Failing that, just enter your passcode and you should be back to normal again.