A myth I didn’t hear about until sometime in the past ten years is that fingernails and hair grow after death. However, I never believed this myth because I heard it in the context of history and how people believed vampires in the past. They used to dig up graves of recently dead people to look for signs of undeath. They apparently often found such signs in the form of what appeared to be hair and nail growth on a corpse. Please note that I personally am not 100% convinced this is the source of the myth because this directly contradicts a similar vampire evidence story (myth?) about people in the Dark Age believing bloated bodies of the recently dead was a sign of a vampire. Not to mention the fact that one of the beliefs about vampires is that if fed with blood, they return to the physical form from the point they become a vampire (meaning their hair can't grow). [How's that for talking on myths about myths? :-)]
The fact is, as I’ve heard from many difference sources by now, the body’s skin dries out sometime after a person dies. As it does so, it shrinks. The nails and hair remain in place, so it can appear as though they grow because they protrude farther out from the skin. That said, I would still guess it is possible the nails and hair do grow a tiny bit right after death, though they certainly do not continue to grow once all body functions stop.