So it turns out the BSODding on the Vaio was more serious, as on a reinstall of XP it would still BSOD towards the end of the install. Thanks to a suggestion from daz71 I tried out Ubuntu, which installed and worked without problem. I was pretty impressed by the Ubuntu install, which has a memory and disk check built into it if need be. I did run a disk check on the laptop without any errors, and am running a memory test now. Ubuntu was a little confusing to begin with, as I couldn't get Flash installed - I was just tinkering around with it, to see how easy it was to use. I found an update tool though, installed lots of updates, and then it worked. I also installed TweetDeck, which worked without issue.
If the memory test is clear I'll try XP again, but if the BSODs continue I'll reinstall Ubuntu and eBay it like that. Not sure how much I'll get for it, but not much I suspect.
Don't seem to be having much luck with hardware at the moment as the ancient PowerBook G4 I borrowed from work won't connect to the wireless network either. Perhaps it can't cope with WPA2 encryption?