In this talk we show how random point sets and associated graphs can be used to approximate the Mumford-Shah functional in the sense of Gamma-convergence. On the one hand, we consider discrete functionals that act on functions defined on stationary stochastic lattices and take into account local interactions through a non-convex potential. In this setting the geometry of the lattice determines the anisotropy of the limit functional. Thus we can use statistically isotropic lattices to approximate the Mumford-Shah functional via stochastic homogenization techniques. On the other hand, a direct finite difference discretization of the Ambrosio-Tortorelli functional yields a convergent approximation, too. For the latter, randomness allows for a lattice spacing proportional to the singular parameter of the continuum approximation. If time permits, we also discuss some numerical aspects of these random methods. This is joint work with Annika Bach and Marco Cicalese.