Found the 2 folders listed below, however both are empty.
I don't understand why I can't get into Safe Boot and Hardware Test. Rebooted from the original install disk (OSX 10.5.2) many times pressing the Shift key or the D key.
What else can I try?
The hardware test (extended testing) passed with "No trouble found". Took over 6 hours to complete. BTW, the reason the D-key did not work turned out to be that I had to plug the keyboard directly into my Mac (and not the monitor).
Had to problem already a few times in Aug 2012 last year on Snow Leopard, but I am on Mountain Lion now. That's why I still think this might be a hardware problem.
Still looking for ideas of what could cause the sudden reboots or how to troubleshoot.