Out of the Blue BSOD

Please remove items from the startup
How to remove items from startup - Windows 7 Help Forums
These programs could interfere with the boot process giving you trouble and increase the boot time.

Unfortunately no dump was created with the last crash, for now we have 2 options.
A memory test,
Driver verifier, if the memory test doesn't show any errors.

Code:

Event[49953]:
  Log Name: System
  Source: Microsoft-Windows-WER-SystemErrorReporting
  Date: 2016-04-05T09:40:23.000
  Event ID: 1005
  Task: N/A
  Level: Error
  Opcode: N/A
  Keyword: Classic
  User: N/A
  User Name: N/A
  Computer: BRENDANE6540
  Description: 
Unable to produce a minidump file from the full dump file.

Diagnostic Test

 RAM TEST

Run MemTest86+ to analyse your RAM. MemTest86+ - Test RAM - Windows 7 Forums

   Note

MemTest86+ needs to be run for at least 8 complete passes for conclusive results. Set it running before you go to bed and leave it overnight. We're looking for zero errors here. Even a single error will indicate RAM failure.

Addition:
If errors show up you can stop the test, remove all sticks but 1 and test this single stick in each slot for 8 passes or until errors show, switch sticks and repeat.
If errors show up and you see them a lot later, no problem, the errors don't affect the test.

Please read this post carefully and don't forget to run MemTest86+ too :)

Thanks for the event data, the 0x80 is a hardware crash what explains why no dump can be created.
Driver verifier can be ignored now, I'll edit my post and remove driver verifier as test.
BSOD Index
The down side of such a crash (hardware crash) is that they usually don't give enough data of what exactly might be the cause, so the difficult part is to find the hardware component that causes it.
A few standard things to try, anything that has already been done? Please try it again and double check to ensure it isn't the cause.

• Roll-back any recent hardware or driver changes;
• Ensure that RAM sticks are matched and undamaged;
• Check for viruses or hard drive damage, infections can cause the system letting think there is a hardware problem;
• Run ChkDsk to identify file system problems;
• Ensure that all connections are sound inside the computer and all cards well-seated

Below are some hardware tests you could try to run.

Diagnostics Test

 HDD TEST

Run SeaTools to check the integrity of your HDD. SeaTools for DOS and Windows - How to Use - Windows 7 Help Forums
Run following tests

1. Short Drive Self Test
2. Short generic
3. Long generic

If the short generic fails, no need for the long generic.

   Note

Do not run SeaTools on an SSD as the results will be invalid.

Run chkdsk
Disk Check - Windows 7 Help Forums

Run HDTune to

• scan for errors, no quick scan but full scan
• check the health,
• benchmark.

It may take some time, but please take the time you need to perform it properly.
Let me know what the results are

• of the error scan,
• make a screenshot of the health of every hard drive and post the screenshots,
• post screenshots with the benchmark of the
  
  • transfer rate,
  • access time,
  • burst rate,
  • cpu usage.

Diagnostic Test

 CPU TEST

Run Prime95 to stress test your CPU. Prime95 - Stress Test Your CPU - Windows 7 Forums

   Warning

Your CPU temperatures will rise quickly while under this stress test. Keep a keen eye on them and abort the test if overheating occurs.

Diagnostic test

 GPU TEST

Run Furmark to test your GPU. Furmark tutorial

   Note

Run Furmark for around 30 minutes

   Warning

Your GPU temperatures will rise quickly while Furmark is running. Keep a keen eye on them and abort the test if temperatures rise too high

Please let me know how it goes, I'm very curious about what the cause is :)