Interestingly I've been abroad with the phone and found that it worked OK when roaming. This leads me to the conclusion that there's issues with the iPhone 5 and certain 3G-only carriers such as '3' in the UK.
Summary of my problems:
Since taking a contract with '3' in the UK, I've suffered from constant dropouts and the phone cycling around dropping the signal, "searching", picking up a cell, switching to 3G, holding that for 30 seconds, then dropping the signal again. My battery is often near exhausted at the end of one day (previous iPhones would last for two days or more).
This dropout cycling does not happen all the time, but is very common when you unlock the phone. Sods law says it never happens if you're in a '3' shop or demoing the problem to anyone.
I've replaced the SIM with no change in symptoms.
No change in iOS has ever changed this.
I have a feeling that this problem didn't beset me when I was with a different carrier.