First, I would check your Device Manager. Is it cleaned up or are there any conflicts? If so, what are they? If it's something you can identify, try to go to the vendor's website and download the latest drivers. Sometimes, conflicts in Device Manager cannot be immediately identified and the unknown (needed driver) will throw itself under 'Other Devices'. That's when it is more difficult (and what happened to me) when I upgraded from Vista to Win 7. I never got a BSOD but new there was an issue in Device Manager and it was determined to be the Intel system device that communicates with my video card. I only mentioned that to you because I saw you have the exact same video card and your dell model is very similar to mine. However, again, it may have nothing to do with your blue screens. However, you may want to read my post about my experience in installing Win 7.
Back to your situation, you could also check event viewer (access that through your search box). Event Viewer will identify error logs on your system. Sometimes that can be a clue as to what is causing a freeze or blue screen. If you use this feature in windows, you only need to focus on the actual error logs, not the warning messages and so forth. There are many things that can do wrong so unfortunately you are going to have to do some investigation. There's no single solution to fit everyone in your situation. I wished I could tell you more.
However, since you are now using a 64 bit operating system, I would deffinately upgrade to at least 4 GB of ram. What you have is quite low for a 64 bit system. If you cannot diagnose the problem, you may want to consider reinstalling the new operating system (hopefully after you've added more ram). If you decide to add more ram, remember physical slots 1 and 3 are a pair and physical slots 2 and 4 are a pair. So you want to keep like ram in the same pairs in order to keep dual channel mode in your memory and to ensure your computer is optimized.
I wished I could give you a more definitive answer but again there's many variables in these situations.
Dell XPS 410
Intel Core 2 Quad @ 2.4 GHz
6.00 GB Memory (Ram) (Dual channel DDR2)
Windows 10 Home (64 bit)
Nvidia GeForce GTX 650 Ti graphics card
LG 16X BD-R 2X BD-RE 16X DVD+R 5X DVD-RAM 12X BD-ROM 4MB Cache SATA Blu-ray Burner
Crucial MX100 256GB SSD