In my time teaching, I have seen many students, of all ages, come through my classroom. The dominant question in the beginning, however, is "How old does a child have to be to start formal music classes?"
... Well, it depends. While I have seen a few exceptionally focused four-year olds who absorb an entire hour's class, that's usually the exception, not the rule. Most very young children do better with a thirty-minute class until they're older. Every child is different, though.
Here are some of the criteria you can use to determine if your child is ready for formal musical training:
1) Does your child know their alphabet?
2) Can your child count to twenty?
3) Can your child add simple, single digit numbers (even if they have to use their fingers)?
4) During playtime, can your child sit still long enough to follow instructions from parents and/or teachers?
5) When given a simple task at home (i.e. "Please put the spoons on the table", or "Please put your cup in the sink"), are they able to carry out the task?
6) Is your child able to play well with others without getting immediately frustrated?
This model is usually a pretty decent barometer to determine if your child is ready to learn formal music. Even if your answer isn't "yes" to all of these right now, I've found that usually, by the age of 5-1/2 to 6, the answer to all of these is almost always "yes".
You can sign them up at any age, but when you're paying for a service, you want your children to learn. On a side note: I remember signing my daughter up for dance when she was three. I thought that she would look so cute in her little ballet flats, leotard, and tights, doing plies... The reality looked more like stampeding herd of elephants (as opposed to the graceful gazelles my mind's eye initially saw), with the little girls running across the stage in ballet flats, leotards, and tights. In fact, the one plie I THINK I saw may have been an accident. And... I think that my daughter was the herd leader. Simply stated, she just wasn't ready for dance, and I wasn't ready for another stage performance where I had to hide in the audience, as my daughter boldly yelled, "HEY, MOMMY! LOOK AT ME!!!" - right before she shook her rear end to everyone in the audience.
... And, now that she's older, I find that ballet was never her cup of tea, anyways. She wanted to do Ju-Jitsu and play football, instead. Age and maturity means a lot when a child begins classes. Who knew? :-)
Music is a beautiful art to learn, but a child will find it much more beautiful if they're ready to learn it.