The conditions under which biological cells switch from a static to a motile state are fundamental to the understanding of many healthy and pathological processes. In this talk, we consider a cell constrained to move along a one-dimensional track. We show that even in the presence of a fully symmetric protrusive activity at the cell edges, such a spontaneous transition can result from a feedback of the deformation of an elastic substrate on the cell traction forces. The loss of symmetry of the traction forces leading to the cell propulsion is rooted in the fact that the surface loading follows the substrate deformation, leading the cell to surf its own wake. The bifurcation between the static and motile states is characterized analytically and, considering the measurements performed on two cell types, we will argue that such an instability can realistically occur on soft in vivo substrates.