While I've seen Emergencies, both as a participant and as a target, I've yet to see an "interesting" Emergency. At least, none that are more interesting than the others, or unexpected.
The potential threat of being the target of an undesirable Emergency has definitely had an effect on my diplomatic efforts, though. In my Georgia game, everybody in the world hated Wilhelmina (which is unfair, she's such a dear sweet lady). Regardless of that, from the time I met her, I made it a point to send her a trade route if I could, and she had several cities within range that offered up good yields, so that made her happy and I believe her secondary agenda was Citystate Protector which worked out quite well since both Gilgamesh and Harald kept getting the same idea in their head: conquer Buenos Aires, which did not work out well for either of them and worked absolute wonders for me. So Wilhelmina and I became good friends, despite my being friends with several of her enemies. (Who doesn't love Tamar, though, tbh.)
Eventually, however, she asked for an alliance and I had to continually turn her down, because she kept getting involved in wars with my other allies. Normally that wouldn't matter, but the AI is wonky as anything when it comes to maintaining friendships and alliances. They must play coy and turn you down for an indeterminate amount of time before they finally accept, or you just have to let them suggest it. During those lapses in our alliance, I couldn't afford (okay, I'll be honest, I could have easily afforded it, I just didn't want to) to risk the Netherlands and my other allies suddenly going to war and dragging me into it, and prompting a Betrayal Emergency with me as the target simply because I was overly nice.
So that was basically the most interesting experience that I've had centered around Emergencies, without actually triggering an Emergency.
