According to the Catholic News Service, "Per capita, Vatican City has the highest crime rate in the world." (read more) I was surprised to hear that. I'm guessing that if one considered violent crime, the ranking would be practically inverted because of the ability to control security there. Then again, who knows. Bottom line: Vatican City is no more heavenly than any other city on earth.
UPDATE: as Ben notes in the comments below, part of the reason for the high crime rate is that there are an enormous number of tourists, which draws pickpockets (who - in turn - contribute disproportionately to the criminal statistics for the burg.