I have so many that I can't choose one to talk about, but a friend of mine had the funniest death.
He was playing AE and was up to Tunnel 7 in Necrem Mines. He tried to sneak past the Slig and when the Slig woke up, my friend panicked. He ran back to the (closed) entrance of that stage and QuickSaved just as the Slig attacked him. So every time the QuickSave loaded, he died.
We spent 10 minutes trying to pause the game at the right moment to Restart Path, because my friend hadn't saved and didn't want to have to do Tunnel 6 again.
EDIT: As for my most embarassing death, I think it would have been in AO and was due to a glitch. It was in the first secret passage in Stockyard Escape. While running away from the Scrab, I noticed that it fell down one of the holes as it was chasing me. Sweet, I thought. This will make things really easy now. WRONG~!
For some reason when I was heading back to get the other 2 Mudokons, the Scrab had re-spawned. IT'S NOT SUPPOSED TO DO THAT! I died miserably, and because I had tried that secret passage so many times in that playthrough, a moving bomb appeared in the foreground without the motion detector being set off. So I was forced to do the whole thing again.
As funny as that death was, it doesn't compare to the more common one; you fall down a hole while riding Elum and you can hear them yelling "aaaaaaaaaaaaaaaaaaaaaaaaaahhhhhh!", but the top of Abe's head is poking through the top of the hole and not moving! Even funnier is when your spirit (that purple stuff that comes out when you die) comes up through the hole and Abe's head slowly moves down as it plays the death sound effect.
(Sorry, it's really hard for me to try and describe that one).