Ice sheets are susceptible to the formation of ice streams, or narrow bands of fast-flowing ice whose high velocities are caused by rapid sliding at the contact between ice and the underlying bed. Based on recent geophysical work which has shown that the sliding motion of ice streams may be described by a Coulomb friction law, we investigate how the location of ice streams depends on the geometry of an ice sheet and on the mechanical properties of the underlying bed. More generally, this problem is relevant to the flow of thin films with Coulomb (or ‘solid’) friction laws applied at their base. By analogy with friction problems in elasticity, we construct a variational formulation for the free boundary between ice streams, where bed failure occurs, and the surrounding ice ridges, where there is little or no sliding. This variational problem takes the form of a non-coercive variational inequality, and we show that solutions exist provided a force and moment balance condition is satisfied. In that case, solutions are also unique except under certain specialized circumstances which are unlikely to arise for a real ice sheet. Further, we show how the variational formulation of the ice flow problem can be exploited to calculate numerical solutions, and to simulate the effect of changing ice geometry and bed friction on the location and velocities of streaming flow. Lastly, we study the effect of ice-shelf buttressing on the flow of ice streams whose spatial extent is determined by the yield stress distribution of the bed. In line with previous studies of ice-shelf buttressing, we find that the removal of an ice shelf can cause an ice stream feeding the ice shelf to speed up considerably, which underlines the important role ice shelves may play in controlling the dynamics of marine ice sheets.