A lot of well known Christmas Carols are in the public domain. This is because they are long out of copyright and nobody owns them. However, although the melody may be out of copyright anyone that does their own arrangement and recording will own the copyright in those elements.
So if you were to arrange and record your own version of a public domain Christmas song you could charge people to use it if you want to.
This applies to any music that is out of copyright including Nursery Rhymes and Classical music too.
The Audio Version