Why does my laptop randomly restart, blue-screen, or work normally on reboot?

"The details are a bit sketchy, but maybe these tips will help.

Make sure your RAM is seated securely.
Make sure your system fans are running as expected.
Try running the laptop just using the power adapter without the battery installed.

If you can find any patterns, they can be a big help in determining the issue. For example, if the machine tends to run fine in the morning and tends to have issues after running for a while, that may indicate a heat related failure.

You mention that sometimes the machine starts normally. When it starts normally, does it run without issue until you attempt to reboot it? When it does run into issues, do you just reboot it until it works correctly, or do you have to turn it off for a while?"

"Need a bit more data before I can give any good advice. In 1), does it get through the bios post before restarting, or does it reboot before it ever tries to start Windows?

Can you try booting with some flavor linux live cd, which may give you a clue to if it's a hardware or software problem? I like to rule out software problems before trying to debug a hardware problem.

Update: Ok, so we can rule out software at this point if it occasionally won't finish post. Because it's a laptop, that makes a few of the common steps impossible.

What I would try is

Reseat your RAM. Depending on the type and number of modules, you could also try with with only one module, then the other, to see if that's part of it.
If you have any modular components such as an ExpressCard or removable wifi module, take it out and see if it does the same thing.
The third would be heat. If this is the culprit, you'll probably find that time of use correlates to the errors you're receiving. Like first thing in the morning it works fine, but after you use it a lot it BSODs. To test this you can use it in a cool place with lots of air flow and see if it makes a difference. Like right in the path of an air conditioning vent."