Preheat the oven to 350°F (175°C). Grease a 9-inch round cake pan and line the bottom with parchment paper.
In a medium bowl, whisk together the all-purpose flour, baking soda, baking powder, and salt. Set aside.
In a large bowl, using an electric mixer, cream the softened unsalted butter and granulated sugar together on medium speed until light and fluffy, about 3 minutes.
Add the egg and vanilla extract to the butter mixture and beat until combined.
Reduce the mixer speed to low and alternate adding the dry flour mixture and the sour cream and milk, beginning and ending with the dry ingredients. Mix until just combined.
Fold in the rainbow sprinkles gently with a spatula.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake for 22-25 minutes, or until the edges are golden and a toothpick inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 10 minutes.
Run a knife around the edges to loosen the cake, then invert it onto a wire rack to cool completely before serving.