Dueling Banjos (comedy/drama): Set in Shelby, North Carolina: a recently deceased man's will forces his two sons, at odds since their mother's untimely death 25 years earlier, to compete against each other in the National Bluegrass Banjo Championship in Kansas, that one of them must win to prevent a priceless family heirloom from ending up as a museum exhibit.

Did you ever see Comfort and Joy by Bill Forsyth who also made Local Hero? It features rivaling ice cream vendors. I'm guessing the story was inspired by the Glasgow Ice Cream Wars, an actual event that took place in the eighties. It started out as a turf war between ice cream vendors and escalated into multiple murders.

pliny and Teddy, that's the kind of thing I was thinking. Some off-beat bit of business that serves as a competition nonetheless. I didn't see it, but I believe "Deck the Halls" did that with rival neighbors and their respective Christmas decorations.