Preheat your oven to 350°F (175°C). Grease and flour a 9-inch round cake pan.
Whisk together the flour, baking powder, baking soda, and salt in a medium bowl. Set aside.
In a large bowl, beat the butter and sugar until light and fluffy, about 3-4 minutes. Add the eggs one at a time, beating well after each addition. Mix in the vanilla extract.
Gradually add the dry ingredients to the butter mixture, alternating with the sour cream, beginning and ending with the dry ingredients. Mix until just combined.
Gently fold in the blackberries, being careful not to overmix.
Pour the batter into the prepared cake pan and spread it evenly. Bake in the preheated oven for 35-40 minutes or until a toothpick inserted into the center of the cake comes out clean.
Allow the cake to cool in the pan for 10 minutes, then transfer to a wire rack to cool completely. If desired, dust with powdered sugar before serving.