According to Metro's Weird fact of the day:
It is against the law to sing off-key in North Carolina.
Is there such a law?
It's unlikely such a law has ever existed, and no record of such a law can be found today. As @Oddthinking has referred to in comments, the closest thing to such a law is North Carolina General Statute 160A-184 which states;
This however, only gives a city the right to make a law regulating the emission of annoying/disturbing/frightening sounds.
The likely origin of this rumor (so called "law") however, pre-dates the statue described above. In 1873, there was a case (State vs. Linkhaw) where a man was disturbing a church service with his singing. The details however, reveal that the problem was not primarily that he was accused of "singing out of tune" but rather that he was disturbing the church service by singing even when the congregation was not.
Although initially he was declared guilty for said disturbance, upon appeal, the courts deemed;
Thank you to Katie Quine of "Our State" who spoke to a state research historian to track down this origin information.