To my knowledge, The Duke never sang in a musical like Clint did in Paint Your Wagon.
Actually John Wayne did play a singiing cowboy in some of those B grade westerms he made in the early '30s.
Having, never seen any of them I can't imagine what they would have been like,
but I distinctly recall reading this in one of his biographies,
because it just seemed such an odd choice having Wayne playing a singing cowboy.

Nor did John Wayne ever battle giant '50s-monsters like Clint Eastwood did in Tarantula
You've got me there.
.