I don't think I've ever seen a shorted LED but I have seen completely open circuit ones. Of all the failures I've seen, at least on 3mm and 5mm types, probably the most common is corrosion of the leads. Mind you, some of my applications involve very harsh environements. Where chemical vapour or liquid contact is likely, I now wipe LED wires with silicone grease then shrink 1.2mm sleeving over them, it seems to be the most effective way of keeping nasties out of reach.
A few years ago, I made a one-off special 'port hole' LED light, it was two concentric rings of 2.5mm copper wire with 36 white 5mm LEDs, each with it's own series resistor wired between them. It made a self-standing structure which was placed in a tube, about 15cm diameter by 15mm deep, on a glass sheet base. It also had a 1N4007 diode wired between the rings for reverse voltage protection and a 3A diode in series with the power cable. In other words, it was very well protected against reverse voltages. After 48 hours of testing at LED maximum current (1A @ 12V) the tube was filled with clear resin, allowed to cure then tested for 48 hours again. The resin, being very similar to the plastic used in the LED body, made the body disappear, making it look as though the glowing junction was floating in a disc of glass. This was intentional for the required appearance and so the light gave a wider beam.
So it has a glass front, the LEDs and all the wiring are encased in deep resin, it has 4 days of testing at full power and is electrically bomb proof. After a few months one of the LEDs went out.....Argggghhhh.
It had to be scrapped and another rebuilt from scratch as there was no way it could be dismantled.
Brian.