We are concerned with the local regularity of solutions to the Stokes and Navier–Stokes equations in the half-space near the boundary. Singular solutions to the Stokes equations are constructed away from the support of non-smooth and localized boundary data or external forces. By employing a perturbative approach, analogous singular solutions can also be established for the Navier–Stokes equations. Furthermore, we investigate the sign behavior of each velocity component in the singular solutions of the Stokes equations, in particular near the boundary, and demonstrate that, in general, reversal flows emerge after the singular time.