As title really, does it over and over again, not just a blip. V frustrating! Any suggestions as to a remedy or should I ask for it to be replaced at a genius appointment?

I think this is an IOS 7 problem. My 5s did it. I calibrated it and it fixed it.

It is not! My iPhone 5 is still on 6.1.4 and has the same issue.

I agree that calibration seems to cure it. I connect the charger briefly to get it to boot, then disconnect the charger and run it down to ~1%. I then give it a charge to 100%+plus a few hours.

Yep as above, my 5 did it a few times but it seemed to sort itself out. But I sometimes leave it on charge for abit even when it's reached 100%, so maybe that sorted it. And it did seem to happen after the ios7 update. All ok now though