This work deals with numerical approximations of unbounded and discontinuous value functions associated to some stochastic control problems. We derive error estimates for monotone schemes based on a Semi-Lagrangian method (or more generally in the form of a Markov chain approximation). A motivation of this study consists in approximating chance-constrained reachability sets. The latters will be characterized as level sets of discontinuous value functions associated to adequate stochastic control problems. A precise analysis of the level-set approach is carried out and some numerical simulations are given to illustrate the approach.