Since at least the late 1700s and into the mid-1800s, it was a crime to steal a shroud from a grave in England. This was not because shrouds were particularly valuable or desirable, but because someone forgot to make it illegal to steal dead bodies. A corpse was officially not property, and therefor could not be stolen. However, they were in some demand as physicians-in-training needed bodies to dissect. For reasons that I have not been able to uncover, the solution to this was not to make the abduction of corpses illegal, but rather to make the theft of any valuables, including the shroud, a crime.
The shroud was chosen, it appears, simply because it was the least valuable thing in the grave, and everyone had one. Until 1814 it was actually illegal to be buried in England without a woolen shroud, and while there may have been some poor black market in these shrouds, there are more references to people complaining that they had to be buried in wool rather than in finer cloths (this was, in fact, exactly why the law was original passed in 1666, to bolster England's wool sales, which were being threatened by the importations of finer cloths).
This bit of legal loophole trivia would have been long forgotten, had it not been listed as one of the 19 Crimes that could get you sent to the penal colonies of Australia. This list makes a point of mentioning petty larceny separately, so if you did not take the shroud but did make off with a ring, you were still in big trouble.
Be warned, grave robbery is now a crime, and you cannot take corpses, shrouded or not.
References:
The Cabinet Lawyer: A Popular Digest of the Laws of England, Civil, Criminal, and Constitutional; Intended for Practical Use and General Information, Longmans, Green, Reader, and Dyer, 1874.
History House: Burried in Wool
Wikipedia: Burying in Woollen Acts