Recovering the 3D shape of deformable surfaces from single images is difficult because many different shapes have very similar projections. This is commonly addressed by restricting the set of possible shapes to linear combinations of deformation modes and by imposing additional geometric constraints. Unfortunately, because image measurements are noisy, such constraints do not always guarantee that the correct shape will be recovered. To overcome this limitation, we introduce an efficient approach to exploring the set of solutions of an objective function based on point-correspondences and to proposing a small set of candidate 3D shapes. This allows the use of additional image information to choose the best one. As a proof of concept, we use either motion or shading cues to this end and show that we can handle a complex objective function without having to solve a difficult non-linear minimization problem. We compare our method against state-of-the-art approaches and demonstrate a marked improvement.