In Stefan type problems, free boundaries may not regularize instantaneously. In particular, there exist examples in which Lipschitz free boundaries preserve corners. Nevertheless, in the two-phase Stefan problem, I. Athanasopoulos, L. Caffarelli, and S. Salsa showed that Lipschitz free boundaries in space-time become smooth under a nondegeneracy condition, as well as sufficiently ""flat"" ones. Their techniques are based on the original work of Caffarelli in the elliptic case. In the talk, we present a more recent approach to investigate the regularity of flat free boundaries for the one-phase Stefan problem. Specifically, it relies on perturbation arguments leading to a linearization of the problem, in the spirit of the elliptic counterpart already developed by D. De Silva. This talk is based on a joint work with D. De Silva and O. Savin.