Need help troubleshooting Dimension E510 startup problem

I am having an intermittent problem starting up my Dell Dimension E510. When I power up the computer, I run into one of the following problems:

1. Sometimes the computer does not even finish POST, and powers off. It tries a few times to automatically restart, but fails to do so.

2. Sometimes the computer finishes booting up into Windows (I am running Windows XP), and before I can login, the "Turn off computer" dialog appears with options to Stand By/Turn Off/Restart. Trying to dismiss the dialog brings up another instance of the same dialog. It is hard to get past this dialog and login.

3. Sometimes, I am able to boot to Windows and login, but the "Turn off computer" dialog appears a few seconds to a few minutes after I login. Trying to dismiss the dialog brings up another instance of the same dialog. When this starts happens, I cannot use the computer other than to turn it off.

In general the computer is pretty much useless. I tried switching keyboards, to no avail. I tried disconnecting all perpherals other than keyboard (connected via USB) and monitor, but that didn't make a difference either.

The troubleshooting LEDs in the front panel are not much help. They blink randomly in the boot sequence, but do not indicate any stable pattern.

It seems that some part in the computer is faulty, but I don't know how to go about isolating it. Any help would be appreciated.

Thanks.