iPhone 5 Wi-Fi Problems

Ever since getting the iPhone 5 I've had problems with its wi-fi. Safari would sometimes work and sometimes not respond at all.

iOS 6.0.1 helped a bit as the wi-fi started to work well with at work, which it previously hadn't, but I'm still having trouble at home. As I say, web pages sometimes load but generally don't. Even trying to access my router via its IP address doesn't always work.

I can see that several others are having trouble with wi-fi, but there's not a huge amount of information around.

On a perhaps related note, my data usage is massively, massively higher than it ever had been, despite my actual usage not changing. I suspect this wi-fi issue is causing it.

I have tried resetting network settings, but it hasn't helped. I've also tried manually setting the DNS on the phone and changing the security on the router, to no avail. I know that the router is not a fault per se as I was using it before - plus have a fourth generation iPad, a MacBook and an iPhone 4S happily using it.

Is anyone experiencing the same? Effectively my iPhone 5 is unusable on my home wi-fi.