In this talk we discuss a variational problem with the goal of approximating a general probability distribution with measures that are uniformly distributed over 1-dimensional (rectifiable) sets. After discussing the initial questions of existence and preliminary qualitative properties, we introduce a phase-field implementation, reminiscent of the work of Ambrosio and Tortorelli for the Mumford-Shah problem, that allows a unified approach to study this problem and the average distance minimization problem.