Yes, the current Ubuntu series (Ubuntu-Unity, Xubuntu-XFCE, Kubuntu-KDE, Mythbuntu-XFCE, Lubuntu-LXDE, Ubuntu Studio-XFCE) is 12.04 Pangolin.
In the recent past, yes, there have been issues with stability with 64 bit on Linux, particularly with 64 bit Flash and related browser issues, plus perhaps some random driver issues with 64 bit though I can't give examples. Just normal laws of computer science in operation (bugs, regressions, etc) as the world is still shifting over to 64 bit, justified or not at this time.
There is no practical, real world benefit to using 64 bit currently for home users other than use of greater than 3-4GB RAM per process. There is a common misunderstanding that 32 bit Linux cannot access more than 4GB RAM. Not true. The limit is 3-4GB per *process*, which means if you have 8GB installed, 4 running apps could use 2GB each, or 8 apps could use 1 GB each, or 2 apps 4GB each. This applies to distros compiled to use PAE, i.e most common/popular distros now:
A thread earlier this year demonstrated that Flash/browser issues are still there for 64bit.
To use the latest Flash on Linux (v11.3.x as of today), you must use the Google Chrome or Chromum (FOSS Chrome) browsers. Firefox is stuck at v11.2.x and may never go higher unless policies change.
Unless you're doing pro-level photo/audio/video editing, just use 32bit for now. By the time you *need* to use 64bit, I suspect there will be no more significant issues being discussed on forums like this re: 64 bit problems, and/or Flash will be banned from the Web