This has been a topic of discussion on the Well Trained Mind board, so I thought I'd address it here.
My first thought is "What songs are important to you"? Every family has their own musical heritage, that springs from their backgrounds, culture, and beliefs (religious and otherwise), and that's really where you start teaching music and sharing music with your children. Often, this comes from daily life, but I've known many parents who, the second a baby is born, stop listening to their music and put Raffi in the CD player. Now, that's not a bad thing-and there are a few genres of music where that may be a good idea. But in most cases, the music you'll be most comfortable singing with your child is the music that you're comfortable with yourself-the songs you sing in the car when they come on the radio, or sing in the shower. And there aren't many genres that don't have at least SOME songs that are child-appropriate. I once had a three year old student who's favorite song was "When a Man loves a Woman". And man, could she sing that song! Nothing wrong with that-it shows that her parents are sharing their music with her, and that their love of music is being passed to her.
Having said that, though, there are certain types of music that I consider important to children.
The first is nursery rhymes/nursery songs. These are crucial for developing phonemic awareness (Mem Fox, in Reading Magic, relates a study done that shows that children who do not enter school with multiple nursery rhymes memorized struggle with reading later). They also are very useful for early singing skills, because in most cases, they're limited in pitch and range to what young children's voices can manage first. Songs written for more adult audiences usually are pitched too low and have too wide of a range for children to sing well. There's a reason why "Mary Had a Little Lamb" and "Ring around the rosey" have stuck around-and they're a place to start.
Recommendation-any good nursery rhyme book. Read, sing, and have fun with your child.
The second category is common heritage songs. In the early 1990's, the Music Educators National conference put together a list of songs, with the help of other music organizations, that they felt were common to the US music heritage. While I may not agree with all their selections (and some that were ranked very high in their surveys were not included due to copyright permissions issues), it's a good place to start.
The Core knowledge folks ("What your nTH grader should know") also have a list of 170 songs they have chosen to include. This list is very diverse, and is absolutely wonderful.
My final category is music from around the world. We include this in our Kindermusik classes and home CDs, but it goes beyond that. My favorite way of finding good world music for children is to go to the source. While Putmayo kids and other world music publishers have done a good job of providing excellent children's music, if you go to an Asian market, or a small grocery store in a Latin American area, and ask what music they recommend for a X yr old child, often they'll point you to resources you've never considered. I am very fortunate in that, teaching at the University of Memphis, I have a lot of parents from different cultures who are often very, very glad to share their music with me, which, in turn, lets me share that with my students, and with you. And if you're traveling, CDs are great souvenirs, light, portable, and easy to bring home.
Regardless, whatever music you choose, have fun, enjoy it, sing, play, and move!