Some of them had Q9650 CPUs (I used one), but I'm not sure which motherboards support them.

Is there a way to find out?

The 960's have jumpers on the fan cable to identify if its copper or aluminum.  PM Chris M about the part numbers for the heatsink.

You have Dell Optiplex 960 MT Desktop Motherboard Y958C 0Y958C

They will fail with unsupported processor on systems without the copper performance heatsink. N385R??So Jumper or not Identify's the fan and Heatsink (This is a guess not fact.)  Bios should be updated to latest BEFORE trying a new cpu. And Antivirus software should not be installed when you try to update the bios.  Bios file should come ONLY from dell not other places like joes crab shack bios.

I suggest you take a careful look at the CPU fan connector. Dell will put a jumper on them to allow a cpu or not.On the version that DOESN'T work with quad cores, the last pin of the header is grounded with a jumper.  Basically does nothing other than to identify it as a different fan.  The version that works with quad cores has the last pin empty and unconnected so not grounded.  Thats the ONLY difference AFAIKT.  When I put the new quad capable heatsink assembly into my original 960 SFF, it now works with quad cores. The mainboard part number between the 2 machines is the same. G261D.

http://www.dell.com/downloads/la/pdf/Optiplex960_Tech_Guidebook_new.pdf

The fan cable connecting to the board has the last hole on the fan cable empty which means it's quad capable right?