The geometry of dissipating motions in direct numerical simulations (DNS) of the incompressible mixing layer is examined. All nine partial derivatives of the velocity field are determined at every grid point in the flow, and various invariants and related quantities are computed from the velocity gradient tensor. Motions characterized by high rates of kinetic energy dissipation and high enstrophy density are of particular interest. Scatter plots of the invariants are mapped out and interesting and unexpected patterns are seen. Depending on initial conditions, each type of shear layer produces its own characteristic scatter plot. In order to provide more detailed information on the distribution of invariants at intermediate and large scales, scatter plots are replaced with more useful number density contour plots. These essentially represent the unnormalized joint probability density function of the two invariants being cross-plotted. Plane mixing layers at the same Reynolds number, but with laminar and turbulent initial conditions, are studied, and comparisons of the rate-of-strain topology of the dissipating motions are made. The results show conclusively that, regardless of initial conditions, the bulk of the total kinetic energy dissipation is contributed by intermediate scale motions, whose local rate-of-strain topology is characterized as unstable-node-saddle-saddle (two positive rate-of-strain eigenvalues, one negative). In addition, it is found that, for these motions, the rate-of-strain invariants tend to approximately follow a straight line relationship, characteristic of a two-dimensional flow with out of plane straining. In contrast, fine-scale motions, which have the highest dissipation, but which only contribute a small fraction of the total dissipation tend toward a fixed ratio of the principal rates of strain.