Unpredictable freezes resulting in a shut down, no logs, dump or BSOD

Hi there, sorry this is my first post and I'm not quite as PC savvy as people would like to think. But this problem has seemed to stump a friend and my uncle, both of whom are more adept with PC's than me.

It started on Wednesday morning I believe, I was nipping on my laptop before I had to go to work to check on emails and things when the system froze. It wasn't a case of the application I was using (Firefox I believe) locking up, everything from system sounds to the cursor wasn't working. I tried a hard reset using the power button and it happened again shortly after it restarted.
After work everything seemed to be fine until later on when I returned to my laptop after leaving it for a few minutes to find it turned off when it had been on. I had a couple more freezes and it turned out that leaving it whilst it was frozen like it was caused the computer to shut itself off with no input from me. However on Thursday, I was able to play on Minecraft for several hours without any problems. I don't believe I had any problems on Thursday at all.
Late Friday night though, the problem returned. It was mostly whilst I was on Firefox once again but after one restart I found that it froze up whilst I was trying to open Skype to check a message from a friend where there were no other programs open.
The frequency of these freezes has been increasing with one yesterday only occurring after I went onto a different tab in Firefox, staying on that one tab for more than 30 minutes didn't do anything whilst most freezes happened within that time frame.
I have done a memory test with no problems and have started trying to back up as many files as I can (although I only have a couple of small USB Flash drives), although the computer seems to work perfectly fine when I run in safe mode with networking. So I've been using that to get as much work as possible done.
I will be trying a reinstall of the OS at some point. I don't have the discs myself so I need to meet with a friend who does, but any advice anyone can provide before then to get me out of Safe Mode would be appreciated.