Single-carrier space-charge-limited (SCL) effects in planar, cylindrical, and spherical samples are considered. A novel approach facilitates approximate analytical solutions to the current transients following application of a voltage step. The SCL transit time varies by only about 15% with geometry, even for large ratios of outer to inner radii. In that case, large initial current spikes occur and there is an absence or weakening of the familiar current cusp. Injection from electrode edges in planar geometry samples can show similar behaviour.