I'm so glad I'm not the only person with this issue!
I have a blue iPhone 5c and the screen randomly pops up on the left-hand edge, and sometimes makes a clicking sound when I lightly press the screen in the middle - basically, when I'm using it normally I can hear the "click" or "pop". As others on here have noted, this is basically the screen pushing back into the plastic casing.
I'm surprised no one else has come to the conclusion I have - this is related to heat. I had the exact same issue with my Nexus 7 tablet, when the device becomes hot internal parts expand slightly, and that seems to cause the screen to rise ever so slightly - which is why it pops back into place when you press on it.
Last night I had been using my 5c and it was hot, and the clicking was so bad that I could repeat the click ten times in a row. However, this morning - when the phone hadn't been used and was "cold" - the clicking is impossible to replicate.
I got my 5c from Carphone Warehouse and not Apple, but I'm tempted to return and ask for a full refund, and show them a print out of this very thread. This appears to be a design problem with the phone itself, I'm not sure it can be rectified. If your 5c isn't making this clicking noise, then it's probably because you got lucky and yours has more glue in the case, or something.
Very shoddy workmanship by Apple - didn't the iPhone 5 also have a "clicking" screen? These phones should be designed to cope with changes in temprature during use.