The molecular requirements and morphology of migrating cells can vary depending on matrix geometry; therefore, predicting the optimal migration strategy or the effect of experimental perturbation is difficult. We present a model of cell motility that encompasses actin-polymerization-based protrusions, actomyosin contractility, variable actin–plasma membrane linkage leading to membrane blebbing, cell–extracellular-matrix adhesion and varying extracellular matrix geometries. This is used to explore the theoretical requirements for rapid migration in different matrix geometries. Confined matrix geometries cause profound shifts in the relationship of adhesion and contractility to cell velocity; indeed, cell–matrix adhesion is dispensable for migration in discontinuous confined environments. The model is challenged to predict the effect of different combinations of kinase inhibitors and integrin depletion in vivo, and in confined matrices based on in vitro two-dimensional measurements. Intravital imaging is used to verify bleb-driven migration at tumour margins, and the predicted response to single and combinatorial manipulations.