Most of my flats are either goatshead stickers or when the valve head irreparably fails (after a long life with no flats).
The most dramatic was when the nice air-conditioned LBS topped off both tires after replacing one flat. About 2 miles down the hot asphalt road, BOOM both tires blew off the rim. No sag was available but it was just a couple of hot miles walking home.
The most unusual time was when I bike-commuted 8 miles to work, then before riding home noticed that my well-worn rear tire was showing threads and bulging over the edge of the rim. The innertube was visible for over 1" through the tear. So I headed out for my commute towards home, riding really slow and absolutely babying it on corners. As an Athena with a 30# bike with a rack and loaded pannier, there was quite a load on the wheels. After making it through the entire 8 mile distance WITHOUT flatting, I took the bike to the LBS to buy a new tire. We were all amazed that you could stick two fingers all the way through the tire split with the innertube undamaged. And most likely the tire had been split on my way to work, so I could have traveled most of the 16 mile round trip with the tire split.