Eukaryotic cells adhere to extracellular matrix during the normal development of the organism, forming static adhesion as well as during cell motility. We study this process by considering a simplified coarse-grained model of a vesicle that has uniform adhesion energy with a flat substrate, mobile-curved membrane proteins and active forces. We find that a high concentration of curved proteins alone increases the spreading of the vesicle, by the self-organization of the curved proteins at the high-curvature vesicle–substrate contact line, thereby reducing the bending energy penalty at the vesicle rim. This is most significant in the regime of low bare vesicle–substrate adhesion. When these curved proteins induce protrusive forces, representing the actin cytoskeleton, we find efficient spreading, in the form of sheet-like lamellipodia. Finally, the same mechanism of spreading is found to include a minimal set of ingredients needed to give rise to motile phenotypes.