Many cell types have the ability to move themselves by crawling on extra-cellular matrices. Although cell motility is governed by actin and myosin filament assembly, the pattern of the movement follows the physical properties of the network ensemble average. The first step of motility, cell spreading on matrix substrates, involves a transition from round cells in suspension to polarized cells on substrates. Here we show that the spreading dynamics on 2D surfaces can be described as a hydrodynamic process. In particular, we show that the transition from isotropic spreading at early time to anisotropic spreading is reminiscent of the fingering instability observed in many spreading fluids. During cell spreading, the main driving force is the polymerization of actin filaments that push the membrane forward. From the equilibrium between the membrane force and the cytoskeleton, we derive a first order expression of the polymerization stress that reproduces the observed behavior. Our model also allows an interpretation of the effects of pharmacological agents altering the polymerization of actin. In particular we describe the influence of Cytochalasin D on the nucleation of the fingering instability.