If I had to guess, I would say that it's possible that the AC adapter is rated too high for your laptop (or perhaps, it's rated correctly, but just pumping out too much voltage). The increased voltage could cause instability issues. Pumping in lower voltage than rated for your laptop would also cause problems, but not if you have a battery installed. Generally when the voltage gets too low (like in brownouts), my laptop falls back to battery and remains stable. But I don't think the laptop can do anything when fed too much voltage except crash. It could also be a faulty battery.
If your laptop is under warranty, ask them to ship you a replacement AC adapter because your old one is dead (most will do this for free without asking for the old one).
You can also try interrupting the ground plug on your laptop by using a 3-to-2 prong adapter. Ground loops are known to cause problems, especially in laptops, audio, and video equipment. To test it, you remove the ground from the equation.
You should also try testing RAM (try running memtest) on battery and on AC. But I still think the problem has something to do with your AC adapter or the battery.
AC plug BSOD 
Linear Mode
