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;
"A city may by ordinance regulate, restrict, or prohibit the production or emission of noises or amplified speech, music, or other sounds that tend to annoy, disturb, or frighten its citizens. (1971, c. 698, s. 1; 1973, c. 426, s. 25.)"
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.
INDICTMENT FOR MISDEMEANOR, TRIED BEFORE RUSSELL, J., AT ROBESON SUPERIOR COURT, SPRING TERM, 1873. DEFENDANT WAS INDICTED FOR DISTURBING A RELIGIOUS CONGREGATION. THE EVIDENCE AS DETAILED BY SEVERAL WITNESSES WAS SUBSTANTIALLY THIS: DEFENDANT IS A MEMBER OF THE METHODIST CHURCH; HE SINGS IN SUCH A WAY AS TO DISTURB THE CONGREGATION; AT THE END OF EACH VERSE, HIS VOICE IS HEARD AFTER ALL THE OTHER SINGERS HAVE CEASED...
Although initially he was declared guilty for said disturbance, upon appeal, the courts deemed;
It would seem that the defendant is a proper subject for the discipline of his church, but not for the discipline of the Courts.
Thank you to Katie Quine of "Our State" who spoke to a state research historian to track down this origin information.