(Revised June 16, 2006)
While there are a few songs for moms, the list of songs for fathers is pretty short. So short, I considered putting Rankin' Don's "Father Goose" songs from Dan Zanes & Friends albums on the list. While fathers are sometimes characters in kids' music (the RTTs' "April 14th," Keith Munslow's "Accidentally"), they're rarely the subject. Here, then, is the short list -- if you have additions, please note them in the comments.
-- "Daddy-O," off Frances England's Fascinating Creatures
-- "My Dad!," off Alex and the Kaleidoscope Band's Kaleidoscope Songs, Vol 1
-- "Cat's in the Cradle," Harry Chapin (just kidding!)
-- "I'm So Glad To Be A Dad," off Dennis Caraher's Bow Wow Baby
-- "My Daddy Is Scratchy," off Jamie Broza's My Daddy Is Scratchy (thanks to Fran for the reminder)
-- "My Daddy (Flies a Ship in the Sky)," off the Daddy-O! Daddy Woody Guthrie tribute
-- "Courtship of Eddie's Father" (see comments)
-- "Dad" by Father Goose (comments)
-- "Thank you, Daddy" (comments)
-- "The Coffee Song," by Ralph's World, off At The Bottom of the Sea. (Yeah, I used it for the mom's list, but a reader reminded me it mentions D...A...D.D.Y.)
A reader also recommended Daddies Sing GoodNight: A Fathers' Collection of Sleepytime Songs, which isn't really a collection of songs about dads, but close enough...
And, a few for the adults (from commenters and e-mailers)...
-- "My Dad (My Pa)" by Nancy Sinatra
-- "Bein' a Dad" by Loudon Wainwright III
-- "The Kids Are Alright" by The Who
-- "Slow Turnin," "Your Dad Did," and "Stolen Moments" - John Hiatt
-- "Still Fighting It," Ben Folds ("You're so much like me / I'm sorry.")