Preheat the oven to 350°F (175°C) and line a 12-cup muffin tin with paper liners.
In a medium bowl, whisk together the all-purpose flour, baking soda, baking powder, salt, and ground cinnamon until well combined.
In a large bowl, combine the granulated sugar, brown sugar, mashed bananas, grated zucchini, grated carrot, vegetable oil, eggs, and vanilla extract. Mix until smooth.
Gradually add the dry ingredients to the wet ingredients and stir 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 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 pan for 5 minutes.
Transfer the muffins to a wire rack to cool completely before serving.