We investigate adaptive anisotropic discretization schemes for hyperbolic systems with stochastic parameters. The methodology is designed in the context of intrusive Galerkin projection methods, with a piecewise polynomial representation at the stochastic level and a finite volume approach in physical space. We design an efficient algorithm to evaluate full upwind matrices in the context of Roe solvers. Moreover, we consider adaptation aiming at selecting the stochastic resolution level with regard to the local smoothness of the solution in the stochastic domain. Numerical results on the stochastic Burgers and Euler equations with shocks are presented. This is joint work with Julie Tryoen (University Paris Est) and Olivier Le Maître (LIMSI-CNRS).