Preheat the oven to 350°F (175°C). Line a 12-cup muffin tin with paper liners or grease the cups.
In a large bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt until well combined.
In a separate medium bowl, mix the mashed bananas, granulated sugar, beaten egg, melted butter, and vanilla extract until smooth.
Pour the banana mixture into the flour mixture and stir gently with a spatula 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 let them cool in the tin for 5 minutes.
Transfer the muffins to a wire rack to cool completely before serving.