We propose a level-set approach to characterize the region occupied by the solid in Stefan problems with and without surface tension, based on their recent probabilistic reformulation. The level-set function is parameterized by a feed-forward neural network, whose parameters are trained using the probabilistic formulation of the Stefan growth condition. The algorithm can handle Stefan problems where the liquid is supercooled and can capture surface tension effects through the simulation of particles along the moving boundary together with an efficient approximation of the mean curvature. We demonstrate the effectiveness of the method on a variety of examples with and without radial symmetry.