I run Windows XP Professional x64 Edition on my 3700+ as well as on my 3400+ Laptop, I have been running it on my Laptop for a few months and on my Desktop for a couple weeks. The only problems I found is obviously drivers, but my mobo's website (ASUS) has all my drivers for x64 and I haven't really personally run into any software problems, than again, I don't use a large variety of software other than Games, some 3D Modeling Programs, VC++ 6, some Converting and Office applications, and so far they all work fine.
I have found a few games which will not work, I know there's probably a list somewhere, but the ones I can't get to work are: Area 51, Doom 3, Splinter Cell 3, and Fahrenheit. Unless you absolutely must play one of those games, I wouldn't necesarily shun against x64, I like it. As for stability question, it's a very stable OS, never had one BSOD, that's probably brought on by the fact that Windows XP Pro x64 is based off of Windows Server 2003.
I'm sure there's alot of other problems with software, but Office apps (I use Office 12 Beta, Office 03 runs fine too) run fine and CD-DVD apps (DivX, Converting Apps, etc.) all work fine for me. I say if you must use XP 32-bit, just Dual-Boot with x64, you'll like it.
sylvez: Windows has "Windows on Windows 64" in x64 edition that runs 32-bit Legacy applications, Windows 32-bit has WoW 32 for 16-bit applications. Only thing is Windows x64 doesn't run any 16-bit applications.