Preheat oven to 375 degrees F (190 degrees C). Grease 12 muffin cups or line with paper muffin liners.
In a medium bowl, whisk together the flour, baking soda, baking powder, cinnamon and salt.
In a large bowl, cream together the butter and sugar until light and fluffy. Beat in the eggs one at a time, then stir in the vanilla. Beat in the flour mixture alternately with the applesauce.
Spoon batter into prepared muffin cups. Bake for 20 to 25 minutes in preheated oven, or until done.