I'm actually running 64-bit kubuntu and it runs fine. Kubuntu and Ubuntu are pretty much the same under the hood so if there are stabality issues it's going to be around the Gnome desktop... as for Kubuntu it's actually working very well for me at the moment.
As someone has mentioned before, there had been issues with Firefox etc... I agree with that.
But, I haven't had any issues with it for some time, it just seems to be working now. I can't recall if I had to tweak some things to get that running smoothly or whether some recent updates have just fixed it.
Oh another issue I did have was with Sound. But again, I did some tweaking at some point and I haven't had issues since. Fresh installs might have cleaned that up too.