I'm curious about this prompt. If something was so embarrassing, why would anyone choose to share it?
As for myself, I can't think of anything that didn't happen while I was still rather young. I've already given my "favorite mistake," when I didn't know what a singletree was.
I would say that everyone has had embarrassing moments in their life, and some may not be able to keep those things from the public eye. For example, can you imagine how embarrassing the railway folks must have been when their train fell from an upper level floor? Many very embarrassing things have been on the news, at one time or another.
"World's Dumbest Criminals" comes to mind. These people are already not the sharpest knife in the drawer, but then, they are shown on TV for all the world to see, as well.
Celebrities also come to mind, with their lifestyles, and personal problems being published for the satisfaction of the public's appetite for more information than they need about the latest laughing stock.