It's been a while since I worked at a computer store, but the heatsink installs I saw...
The most common one was people forgetting to remove the little plastic bit over the thermal paste.
Second most common was actually forgetting to mount it. I guess they thought they just needed to set it on there? Either way, it came in just hanging off it's power cord, no where near the CPU.
We had one guy mount his heatsink with drywall screws into the back-plate... through the motherboard. Yeah, that must have been a light show when he powered it on... really surprised he didn't get electrocuted considering he'd grounded it to the case.
Expanding foam holding it in place. In the process of removing it, it pulled several components off the board, ruining it.
One guy couldn't figure out the mounting clamps, but knew better than to just set it there, so he hot glued it down.
Then there were the ones that came in because the computer wasn't running right, and we'd pull off the cover to find the fans had completely seized and the heatsink looked more like a dust bunny. You could tell which ones came from houses where the users smoked, because the heatsinks would be coated in this... brownish sticky goop under all the dirt and were really gross.