Even try out 10.2 if you can't afford any of the others. Mac OS X 10.2 was even still much better than aything before it. I've used it on some older G4 Power Macs and they were still quite a joy to use, as well as being well supported by software developers.
Elander is correct in his assessment that you are encountering kernel panics. KPs or most often the result of hardware problems and can be caused by attached devices but most often are the result of marginal RAM. Software tests of RAM may or may not be able to actually detect marginal RAM but it is certainly worth a try. Rember is a GUI interface to a command line memory test utility. It can take a long time to run one pass and to have any hope of finding errors you generally have to run it for several iterations.
The only completely reliable way to test RAM is to successively pull DIMMs and see if the problem goes away. That may mean having to run for an extended period of time to be sure when you have found a problematic DIMM.
Contrary to what others have said, in my experience, if the problem is hardware related upgrading the OS will aggravate the problem rather than offering any relief.