I was watching Barney the other day over at a friend's house that has little ones, and I got to thinking about the copyright over these songs : what is the deal? Are the songs over 50 years old copyright free?
What about the actual nursery rhyme consider they are probably easily over 100 years old for some of them...?

Copyright is always going to be a case by case basis.

And remember, it's not just the "song" that may be protected - a performance, a recording, the songwriting, the arranging, there are a LOT of creative sub-parts to a particular performance that may be protected as well.

One good resource I've used over the years is BZ Rights and Permissions in NYC. They publish a directory with a lot of information on what "familiar" works are (or are not) still under copyright.

(Surprised me to discover that "Happy Birthday" is a copyright protected work! which is one reason why so many chain restaurants subject us to their own "CUSTOM" happy birthday annoyance!)

That's where I'd start to backtrack rights to a nursery rhyme.

The only way to be SAFE when using something you didn't totally create yourself is to talk to an appropriate professional who knows the law involved.

The stickied thread: Fair use copyright law: The Comic Book!
should answer some of your questions.

Interestingly enough (as I understand it) Happy Birthday is copyight protected, but "Good morning dear teacher" isn't.

The only difference is the length of the second note (the "py" of Happy)

In Happy Birthday it's a dotted eighth followed by a 16th note, in "Good morning..." it's two eighth notes.

In other words "ha--- Py" qs opposed to "ha-py-"

We always used arrangement of Good Morning Dear Teacher and just had everybody play the first lote long and the second note short - works fine.