Preheat your oven to 350°F (175°C). Grease and flour a 9-inch round cake pan or line it with parchment paper.
In a medium bowl, whisk together the flour, baking powder, baking soda, and salt. Set aside.
In a large mixing bowl, cream the softened butter and granulated sugar together until light and fluffy, about 3-4 minutes.
Add the eggs one at a time, mixing well after each addition. Stir in the vanilla extract.
Gradually add the dry ingredients to the wet mixture, alternating with the sour cream. Begin and end with the flour mixture, mixing until just combined.
Gently fold in the mulberries, being careful not to break them apart.
Pour the batter into the prepared cake pan and spread it evenly.
Bake for 30-35 minutes, or until a toothpick inserted into the center comes out clean. Let the cake cool in the pan for 10 minutes before transferring it to a wire rack to cool completely.
Once cooled, dust the top with powdered sugar if desired before serving.