Substrate-mediated fusion of small polymersomes, derived from mixtures of lipids and amphiphilic block copolymers, produces hybrid, supported planar bilayers at hydrophilic surfaces, monolayers at hydrophobic surfaces, and binary monolayer/bilayer patterns at amphiphilic surfaces, directly responding to local measures of (and variations in) surface free energy. Despite the large thickness mismatch in their hydrophobic cores, the hybrid membranes do not exhibit microscopic phase separation, reflecting irreversible adsorption and limited lateral reorganization of the polymer component. With increasing fluid-phase lipid fraction, these hybrid, supported membranes undergo a fluidity transition, producing a fully percolating fluid lipid phase beyond a critical area fraction, which matches the percolation threshold for the immobile point obstacles. This then suggests that polymer-lipid hybrid membranes might be useful models for studying obstructed diffusion, such as occurs in lipid membranes containing proteins.