We consider a singular stochastic control problem with a radially symmetric running cost. We show that the value function is smooth, the nonaction region is a ball, and the problem has an explicit solution in terms of power series. Also, for a singular ergodic control problem with the class of admissible processes constrained to Brownian motions reflected normally at the boundary of some open, connected Caccioppoli set, we show existence, regularity, and basic properties of optimal domains using a geometric measure-theoretic approach.