Hmm - Great question CBL
Countless times it has done its job, fixed things and has been a 'godsend' - all the normal situations regarding screws, cutting, trimming (I hate snaged fingernails), sawed down branches, opening cans when no-one has a can opener etc etc!!
So maybe it has 'saved the day' - But sadly I don't think it has ever really 'saved my bacon'.
Closest I got was student parties (when typically the domiciles are under-equipped) and of course there was never a corkscrew for the botles of wine - except on my SAK!! But hardly a critical situation.
Does remind me of one story though: 'Touching the Void'.
Simon Yates was lowering his badly injured climbing partner down off a mountain called Suila Grande in South America somewhere. Joe went over the edge of a crevasse and was hanging in 'thin air' - Simon was being pulled out of his snow 'bucket seat' (which he was using to belay) by Joe's weight on the rope to which Simon was attached. He cut the rope with his SAK which caused Joe to fall, and almost certainly saved Simon's life. Joe landed on a ledge.
This caused much controversy in climbing circles at the time, but Joe to this day credits Simon with saving his life, and made this very clear in his book which came out a few years later. If you have not read the book it is throughly recommended and the film is pretty good too.