Here are three examples given from Merrian Webster's definition of pardox, there are more there:
Mr. Guppy propounds for Mr. Smallweed's consideration the paradox that the more you drink the thirstier you are and reclines his head upon the window-sill in a state of hopeless languor. —Charles Dickens, Bleak House, 1852-53
It is a paradox that computers need maintenance so often, since they are meant to save people time.
As an actor, he's a paradox—he loves being in the public eye but also deeply values and protects his privacy.
As for songs, I can't think of any offhand, but I am sure there are some.