Many PDE driven inverse problems are notoriously ill-posed. In this talk I will discuss three robust mechanisms leading to instability. These are based on strong global, weak global and only microlocal smoothing properties of the associated forward operators. As a consequence, these mechanism are applicable to a wide range of inverse problems including elliptic, parabolic and hyperbolic ones. The talk is based on joint work with Herbert Koch (U. Bonn) and Mikko Salo (U. Jyväskylä).