I've had a few failures large enough to be memorable, which curiously have all happened in the last year:

The cleat on my shoe decided to shear off, leaving the cleat firmly embedded in the pedal. None of the tools I had with me could produce enough torque to remove it from the pedal. The road shoe was pretty slick on the pedal, which basically meant I had to ride the last 5 miles home with one leg.

I broke a chain last winter on my commute home and didn't have the tools with me to fix it. Even when riding in sub-zero temps I don't wear anything more than a t-shirt and a windbreaker, as otherwise I overheat and sweat profusely. However, I discovered this amount of covering is inadequate when walking at -6 F. I was pretty frozen after three miles of walking when I finally reached the visitor center of a local landmark. I was able to use their phone and call my wife for a rescue, then wait in the heated area. It was after this little stunt that my wife finally insisted that I get a cell phone.

My most recent incident was about a month ago. The tube in my rear tire exploded catestrophically. Sounded like a gun shot. I replaced the tube and continued to ride when it happened again. Both tubes had splits in them about 6 inches long, so there was no patching them. I discovered the tire had developed a split in the sidewall that would rupture the tube. I was out of spare tubes, so I used my (now) 6 month old cell phone to call my wife for another rescue.