Shear flow over a plane wall that contains an axisymmetric depression or pore is studied using a new boundary integral method which is suitable for computing three‐dimensional Stokes flow within axisymmetric domains. Numerical results are presented for cavities in the shape of a section of a sphere or a circular cylinder of finite length, and for a family of pores or orifices with finite thickness. The results illustrate the distribution of shear stresses over the plane wall and inside the cavities or pores. It is found that in most cases, the distribution of shear stresses over the plane wall, around the depressions, is well approximated with that for flow over an orifice of infinitesimal thickness for which an exact solution is available. The kinematic structure of the flow is discussed with reference to eddy formation and three‐dimensional flow reversal. It is shown that the thickness of a circular orifice or depth of a pore play an important role in determining the kinematical structure of the flow underneath the orifice in the lower half‐space.