Nonlinear Grassmannians, i.e., manifolds of submanifolds, can be used to describe certain coadjoint orbits of diffeomorphism groups, including the group of all diffeomorphisms, the group of volume preserving diffeomorphisms, the group of symplectomorphisms, and the group of contact diffeomorphisms. Depending on the geometric structure (volume form, symplectic form, contact hyperplane field) the nonlinear Grassmannians need to be augmented by various additional data on the submanifold. In this talk we will survey the augmented nonlinear Grassmannians appearing in the description of coadjoint orbits, placing emphasis on their smooth structure. These descriptions of coadjoint orbits are useful when constructing singular solutions of the geodesic equation on diffeomorphism groups. This talk is based on joint work with Cornelia Vizman.