This issue is typically related to hardware failure, however did you get these problems before upgrading to Windows 7?
If the answer is no, can you downgrade to Vista or XP and see if you still get it.
If you no longer get it, it was most likely a driver issue... check for updates and/or go straight to hardware manufacturer sites for better drivers.
do you have any additional weird or unusual USB gadgets or add on cards? I have seen everything from someone using one of those USB battery chargers with incorrect drivers causing a BSOD (driver issue again). Try unplugging everything other than the bare essentials and see if you still get it.
Try installing Process Explorer and keep it running. Next time this happens, go to it and click on any of the three charts at the top. A historical graph window will pop up and you can drag your mouse over any of the peaks to see what was causing it. - just in case something is at 100%
Can you try starting in safe mode and seeing if you get it?
If you never got this in your previous OS, it may just be software / driver related, however all your symptoms relate to hardware failure (typically seen on machines with failing hard drives or blown capacitors.)
Blown capacitors -
The top should be almost flat (with slight indents out embossed sections depending on specification... look at the middle one) but you do not want to see any big bumps such as the first one or any leakage as the last one.
If your PC is of the P4 generation, there were MANY MANY boards that had this problem which lead to a lot of random failures as you are experiencing now. I had this problem frequently after just 6 months of heavy usage.