Preheat the oven to 350°F (175°C). Line a 12-cup muffin tin with paper liners or grease the cups.
In a medium bowl, whisk together the all-purpose flour, baking soda, baking powder, and salt. Set aside.
In a large bowl, combine the granulated sugar, light brown sugar, mashed bananas, sour cream, vegetable oil, eggs, and vanilla extract. Mix until smooth and well combined.
Gradually add the dry ingredients to the wet ingredients, folding gently with a spatula until just combined. Do not overmix; the batter should be slightly lumpy.
Divide the batter evenly among the prepared muffin cups, filling each about 3/4 full.
Bake in the preheated oven for 18 to 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, then transfer to a wire rack to cool completely.