Preheat the oven to 350°F (175°C). Line a 12-cup muffin tin with paper liners.
In a large bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt.
In a separate medium bowl, combine the mashed bananas, granulated sugar, beaten egg, melted butter, and vanilla extract. Mix well until smooth.
Pour the banana mixture into the dry ingredients and stir gently until just combined. Do not overmix; the batter should be slightly lumpy.
Divide the batter evenly among the 12 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 tin for 5 minutes. Then transfer the muffins to a wire rack to cool completely before serving.