I seem to recall Chocobo racing was originally a sport for Ul'dahn elites up until Manderville brought it over to the GS. I guess the same could be said about many of the minigames there, too.

Fist-fighting and betting the winner seems to be pretty popular amongst commoners

The Platinum Mirage is home to gambling halls, so we can only speculate what kind of gambling games they do there

The Gladiators in the Blood Sands is one of the biggest spectator sports in Eorzea

I think there was a Chocobo fighting ring in Ishgard