How old does someone have to be before them singing a copyrighted song on stage is illegal?

Age is not generally a consideration. The question would be the purpose of the performance, who owns the copyright, what license you have (if any), among other things. Not all public performances of copyrighted songs are required to be licensed by the copyright owners to the performers. The venue may be a member of a licensed performance rights company, or otherwise have the necessary licenses for ANY performances of the cataloged works.

The only major issue of "age" would be whether the performer were old enough to be liable as an adult or if their parents or guardians would be the ones sued, assuming there was any proof of an actual copyright infringement.

There is no age limit

Most people will not enforce the copyright unless the person is signing it to earn money

Parents (and other adults responsible for children's behavior) can get sued for allowing kids to sing copyrighted songs. It has actually happened a few times when administrators have failed to secure the rights to perform a show at school or a community theater event, but those suits were brought by publishing companies like BMI and ASCAP. In most cases, the negative publicity of suing children (even through their parents) would deter most artists from suing them directly. The answer to your question is that there is no age below which it is okay to sing someone else's song without permission. There is just an age below which the kids themselves will not get sued and, instead, their parents or teachers might get sued.

Nobody has to sign anything for a song to get copyright protection and that protection is not limited to adults. Kids or adults can sing a protected song (virtually all recently written songs) without permission as long as it is not done commercially (comercially recorded for profit) or performed where a fee is charged.

It is illegal to sing a copyrighted song at any age.

There is no age.

Age has nothing to do with it. Copywrite and royalties do.

if their being paid to sing, there is no cut off age.

it's not age -- it's being paid or otherwise compensated.

i'm not aware of any age guidelines, but in most cases the venue pays for the right to perform covers as part of their ASCAP dues and the artists don't have to worry about it.