Partial differential equations which are derivable from minimization principles can often be considered as geodesic equations in infinite-dimensional spaces. (Important examples include the Euler equation for ideal fluids and the Korteweg-deVries equation.) When the PDE is well-posed, the map from initial data to the solution is a smooth Riemannian exponential map. As such, we can consider its singularities. Fredholmness of the exponential map implies that the singularities are all effectively finite-dimensional, which gives us information about the structure of the configuration space. Such techniques can be used even for nonphysical equations, such as geodesics on the group of symplectic diffeomorphisms or contact diffeomorphisms, to get information about the topology of the configuration space. In this talk I will demonstrate a simple and effective criterion for Fredholmness when the geodesics arise from a right-invariant Riemannian metric on a diffeomorphism group. I will discuss how this applies to the Euler equation (on the volumorphism group), the KdV and Camassa-Holm equations (on the Bott-Virasoro group), and the equations obtained for symplectomorphisms and contactomorphisms. I will also discuss how things change depending on the Sobolev norms one uses to get the topology on the diffeomorphism group and the Riemannian metric.