Configure the system to collect small memory dump (minidumps) following:
Can you describe the noise the laptop makes? Beeps?
Upload a screenshot of your hard disk using CrystalDiskInfo and Summary tab using Speccy:
Make a hard drive test from the hard drive manufacturers website:
Number down items at start-up. Keep only your anti-virus, this also improves the time for logging into windows.
AVG contributes in BSOD's, Please remove it using its removal tool from here: Uninstallers (removal tools) for common antivirus software - ESET Knowledgebase
We recommended Microsoft Security Essentials from a strict BSOD perspective. Is most compatible & provides most stability compared to other antivirus app/Internet Security Software.
Do not start the free trial of Malware Bytes; remember to deselect that option when prompted.
Run a full scan with both (separately) once downloaded, installed and updated.
Check your hard drive(s) for errors by running chkdsk
Let us know the results.