We begin with an overview of some aspects of mechanics on Lie groups that go back to Poincaré and Arnold, including the geometric formulation the Euler and the averaged Euler equations in fluid mechanics. We discuss the EPDiff equations (Euler-Poincaré for the Diffeomorphism group), which are PDE's in n-dimensions that reduce to the integrable shallow water equation studied by Camassa and Holm in one dimension and which agree with the Averaged Template Matching Equations from computer vision in higher dimensions. We will be particularly interested in the geometry and dynamics of singular solutions of these equations, which correspond to point vortices, vortex lines and sheets in fluid mechanics. Momentum maps will be shown to be critical to the understanding of these solutions. Simulations (due to Martin Staley) will be presented.