We investigate the mechanics of ice streams and glaciers flowing over a bed consisting of Coulomb-plastic subglacial sediment, or more generally, of channel flows with Coulomb or ‘solid’ friction laws at the boundary. Sliding is assumed to occur if shear stress at the glacier bed attains a prescribed, locally defined yield stress, while no sliding is assumed possible below that yield stress. Importantly, the location of regions of slip and no slip at the bed is not known a priori, but forms part of the solution. By analogy with friction problems in elasticity, we derive a weak formulation as a semi-coercive variational inequality, which admits a unique solution provided a solvability condition ensuring force balance is satisfied. The variational formulation is then exploited to calculate numerical solutions, and we investigate the effect of variations in subglacial water pressure, ice thickness and surface slope on the discharge of a valley glacier with a plastic bed. Significant differences are found between the behaviour of wide and narrow as well as steep and shallow-angled glaciers, and our results further indicate the need to develop models capable of accounting for longitudinal stresses.