In this talk, we consider the Forward--Backward splitting (a.k.a. proximal/projected gradient) algorithm and its variants (inertial schemes, FISTA) for solving structured optimization problem. The goal of this talk is to establish the local convergence of these methods when the involved functions are partly smooth relative to an active manifold. We show that all these methods correctly identify the active manifolds in finite time, and then enter a local linear convergence regime, which is characterize precisely based on the geometry of the underlying smooth manifold. The obtained result is verified by several concrete numerical experiments arising from compressed sensing, signal/image processing and machine learning.