Preheat the oven to 350°F (175°C). Line a 12-cup muffin tin with paper liners or lightly grease the cups.
In a medium bowl, whisk together the all-purpose flour, baking soda, baking powder, salt, ground cinnamon, and ground nutmeg until well combined.
In a large bowl, combine the mashed bananas, unsweetened applesauce, granulated sugar, brown sugar, vegetable oil, eggs, and vanilla extract. Whisk until smooth and fully incorporated.
Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined. Do not overmix.
Fold in the shredded zucchini until evenly distributed throughout the batter.
Divide the batter evenly among the prepared muffin cups, filling each about 3/4 full.
Bake for 18 to 22 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.