We use a numerical, nonlinear, multigrid magnetic relaxation technique to investigate the generation of current sheets in three-dimensional magnetic flux braiding experiments. We are able to catalog the relaxed nonlinear force-free equilibria resulting from the application of deformations to an initially undisturbed region of plasma containing a uniform vertical magnetic field. The deformations are manifested by imposing motions on the bounding planes to which the magnetic field is anchored. Once imposed, the new distribution of magnetic footpoints are then taken to be fixed, so that the rest of the plasma must then relax to a new equilibrium configuration. For the class of footpoint motions that we have examined, we find that singular and nonsingular equilibria can be generated. By singular we mean that within the limits imposed by numerical resolution, we find that there is no convergence to a well-defined equilibrium as the number of grid points in the numerical domain is increased. These singular equilibria contain current “sheets” of ever increasing current intensity and decreasing width; they occur when the footpoint motions exceed a certain threshold and must include both twist and shear in order to be effective. On the basis of these results, we contend that flux braiding will indeed result in significant current generation. We discuss the implications of our results for coronal heating.