Typically 32 bit software will run in a 64 bit OS. 64 bit software won't run in a 32 bit OS. The problem software not running in a 64 bit OS is legacy 1y6 bit applications (not 32). I have Foxpro for DOS 2.6 (16 bit) which will run is 32 bit XP but not 64 bit Windows 7. But 32 bit version of Foxpro do run in Windows 7 64 bit.
Your wording of the question is bad. Basically all processors are 64 bit. A 32 or 64 bit OS will run on a 64 bit processor but the 32 bit will have access to less resources. 64 bit OS's are backward compatible but 32 bit isn't forward compatible. So when software is written it's either compiled into 32 or 64 bit (most 64 now) and has access to more resources that OS provides.
So software isn't really...