Preheat oven to 350 degrees F. Lightly grease 12 muffin cups, or line with muffin papers.
In a large bowl, mix together 1 3/4 cups flour, baking soda, baking powder and salt.
In another bowl, beat together bananas, buttermilk, sugars, egg, vanilla, melted butter and Rum. Stir the banana mixture into the flour mixture just until moistened. Spoon batter into prepared muffin cups.
Bake in preheated oven for 18-22 minutes, until a toothpick inserted into center of a muffin comes out clean.