Preheat the oven to 350°F (175°C). Line a 12-cup muffin tin with paper liners or grease the cups lightly.
In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt. Set aside.
In a large bowl, combine the granulated sugar and melted butter. Stir until well mixed.
Add the eggs one at a time to the butter and sugar mixture, beating well after each addition.
Stir in the vanilla extract.
Mix in the mashed bananas and sour cream until fully combined.
Gradually add the dry ingredients to the wet ingredients, folding gently until just combined. Do not overmix.
Divide the batter evenly among the 12 muffin cups, filling each about 3/4 full.
Bake in the preheated oven for 18-20 minutes, or until a toothpick inserted into the center of a muffin comes out clean.
Remove the muffins from the oven and allow them to cool in the pan for 5 minutes.
Transfer the muffins to a wire rack to cool completely before serving.