Self-propelled particles such as bacteria or algae swimming through a fluid are non-equilibrium systems where particle motility breaks microscopic detailed balance, often resulting in large-scale collective motion. Previous theoretical work has identified long-ranged hydrodynamic interactions as the driver of collective motion in unbounded suspensions of rear-actuated ("pusher") microswimmers. In contrast, most experimental studies of collective motion in microswimmer suspensions have been carried out in restricted geometries where both the swimmers' motion and their long-range flow fields become altered due to the proximity of a boundary. Here, we study numerically a minimal model of microswimmers in such a restricted geometry, where the particles move in the midplane between two no-slip walls. For pushers, we demonstrate collective motion with short-ranged order, in contrast with the long-ranged flows observed in unbounded systems. For front-actuated ("puller") microswimmers, we discover a long-wavelength density instability resulting in the formation of dense microswimmer clusters. Both types of collective motion are fundamentally different from their previously studied counterparts in unbounded domains. Our results show that this difference is dictated by the geometrical restriction of the swimmers' motion, while hydrodynamic screening due to the presence of a wall is subdominant in determining the suspension's collective state.|Understanding the mechanisms that shape collective swimming of microorganisms is of great interest in biology, ecology and physics. Here the authors show that geometric constraints on the swimmers' dynamics, such as near a solid surface, significantly alter emergent collective patterns, with relevance to many experimental and biological microswimmer realisations.