Unstable dimension variability (UDV) is an extreme form of non-hyperbolic behavior, characterized by a violation of the continuous splitting of the tangent space into invariant subspaces. If UDV occurs in a dynamical system, there may be severe problems of shadowability of computer-generated orbits, since the shadowability time is usually small. Riddled basins of attraction have characteristic scaling properties that can be studied by means of a stochastic model for the finite-time Lyapunov exponents. In this work we investigate the presence of UDV in coupled systems exhibiting synchronized states, and its relation with riddled basins of chaotic synchronization. We make this investigation for both discrete-time and continuous-time systems.