My little ones are growing up, but I sang all manner of songs to them when they were younger. Sweet Baby James was a favorite bedtime song for my son. I sang lots of innocuous older pop songs to my daughter -- The End of the World (not REM, but the old "why does the sun go on shining?" one), and the other one from around the same time period that goes "I've told every little star/just how great I think you are/why haven't I told you? Ba da, ba da da da da da da...". I was always pleased that my daughter's favorite was one I had written, though; score one for Dad! (She's 17 now, and doesn't ask for it so much any more.)
It doesn't matter what you sing to your kids as long as you DO sing to them.