I became a single father of 3 year old twins and if they (now 38) were to answer this question I'm pretty sure what some of the songs would be. While driving in our old Dodge truck, they would say, "Sing that Blow Up Your TV song, Papa." They also seemed to like Mama, Don't Let Your Babies Grow Up To Be Cowboys.
Once at a festival, I sang an old Gene Autry song that I learned as a kid called I'm A Cowpoke Pokin' Along. One of my kids was in the audience and he poked his wife and said, "Papa used to sing that while he bounced us on his knees."