I have included this slightly gruesome story under 'fun' rather than 'health' as it isn't a matter of practical concern, just curiosity.
It's often said that human fingernails and hair continue to grow after death, allowing artists who specialise in the grotesque to envisage strangely hairy and long-nailed corpses. In a sense 'grow' is the wrong word for these extremeties made up of a tough, structural protein called keratin (skin, hooves, horns and beaks are also made of different assemblies of keratin). This is because they aren't alive, so can't strictly grow, but rather are constructed by specialist cells in the body.
If we get past the word, though, the idea itself has no basis. The processes that produce hair and nails require a living body to provide the raw materials and to power the production with glucose. Although there aren't any detailed scientific studies testing for this, there is no reason to think that hair and nails would continue to grow, as there is no mechanism to 'power' their production once the body has died. It's a myth.
It has been suggested that the myth originated because skin contracts as it dries and may seem to make hair and nails a little longer a little while after death. According to the British Medical Journal, the myth has certainly been around since the late 1920s, when it was mentioned in the novel All Quiet on the Western Front.