We investigate the motion of a suspended non-Brownian sphere past a fixed cylindrical or spherical obstacle in the limit of zero Reynolds number for arbitrary particle–obstacle aspect ratios. We consider both a suspended sphere moving in a quiescent fluid under the action of a uniform force as well as a uniform ambient velocity field driving a freely suspended particle. We determine the distribution of particles around a single obstacle and solve for the individual particle trajectories to comment on the transport of dilute suspensions past an array of fixed obstacles. First, we obtain an expression for the probability density function governing the distribution of a dilute suspension of particles around an isolated obstacle, and we show that it is isotropic. We then present an analytical expression – derived using both Eulerian and Lagrangian approaches – for the minimum particle–obstacle separation attained during the motion, as a function of the incoming impact parameter, i.e. the initial offset between the line of motion far from the obstacle and a parallel line that goes through its centre. Further, we derive the asymptotic behaviour for small initial offsets and show that the minimum separation decays exponentially. Finally we use this analytical expression to define an effective hydrodynamic surface roughness based on the net lateral displacement experienced by a suspended sphere moving past an obstacle.