They have to have happened during gameplay. They can be funny, weird, badass, or whatever. Try and avoid anything story related, as that would be spoiling it for those of us who haven't finished the game yet.
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Mine's more a glitch than a badass story, really. I don't know if this has happened to other people at all. I was doing a mission at OKB, a side op, where I had to sneak in and rescue two prisoners. So I grab Quiet and we head out. She's got a silenced sniper and I'm still using the Fulton balloons, waiting on my R&D team to finish my wormhole device. So I have to be super careful. First two gates, I knock everyone out, send them back to base. Quiet is standing by, marking guards. I get to the third gate, knock everyone out and extract them and prisoner one. Just as I am extracting him, another guard shows up out of nowhere that neither Quiet or I had seen. Alert phase. I'm out of ammo, she's doing God knows what, and he's in a corner, so I run for it (stupidly). I think I was so surprised to see someone there that I just panicked. Guards start streaming out of the elevator area, so I go back to gate two to extract the other prisoner and get out. He's in a locked room. Guards are now in a reduced alert, and I can hear Quiet picking them off one by one. So I grab the target and head for the door, but... there's a noise outside. I drop him, open the door... Outside there are 15-20 identical guards, all moving in unison, looking for me. They spot me and my screen is a forest of !!! So I decide to go ham and beat them all down. Problem is, they instantly respawn! As soon as I knock one out, another one shows up in his place. Finally it was too much for me and they shot me down. Mission failed. Hasn't happened since, and I've replayed that side op twice.