In this paper, a simple shear flow in a half-space, which has interesting properties from the point of view of boundary regularity, is described. It is a solution with a bounded velocity field to both the homogeneous Stokes system and the Navier-Stokes equation, and satisfies the homogeneous initial and boundary conditions. The gradient of the solution may become unbounded near the boundary. The example significantly simplifies an earlier construction by K. Kang, and shows that the boundary estimates obtained in a recent paper by the first author are sharp. Bibliography: 4 titles.