Having covered thoroughbred racing for 45 years the song that comes to mind is "Fugue for Tinhorns ("I got the horse right here...") from Guys and Dolls.
But horse racing is better known for sayings and poems than songs:
One white foot, ride him for your life.
Two white feet, give him to your wife.
Three white feet, keep him not a day.
Four white feet, send him far away.
Of course, Secretariat had three white stockings so I wouldn't put too much stock in that one.
Racing is a business and it takes a lot of jack,
To train and raise your horses but the way to get it back,
Is by putting over longshots with a nice, big, juicy bet,
But when you spread the word around, short odds is what you get.
So why should a guy who spends his time to train and race his plugs.
Go and tell a hot tip to a bunch of racetrack mugs?
--Ken Kling
"There are a hundred ways to lose a race."
Yeah, and the horse I bet knew every one of them."
"That horse is so slow he's just wearing horseshoes for luck."
"The horse I bet went off at 25/1.
Unfortunately he came home at ten after two."