The rational blowdown operation in 4-manifold topology replaces a neighborhood of a configuration of spheres by a rational homology ball. This is a symplectic surgery that is often used to create exotic 4-manifolds. Configurations of spheres that can be rationally blown down typically arise from resolutions of surface singularities that admit rational homology disk smoothings. It is an interesting question to describe all such singularities, and also to determine whether the corresponding contact 3-manifolds (links) can have  QHD symplectic fillings without admitting QHD smoothings. Previously known results (Stipsicz-Szabo-Wahl and Bhupal-Stipsicz) give restrictive necessary conditions that rule out existence of both QHD smoothings and QHD symplectic fillings in many cases, by topological and symplectic methods. However, we find examples of unexpected rational blowdowns, where QHD Stein fillings exists but QHD smoothings don't.  Inspiration for this contrast between algebro-geometric and symplectic pictures comes from de Jong-van Straten’s description of Milnor fibers of sandwiched singularities and a symplectic analog of their theory that we develop. Joint with Marton Beke and Laura Starkston.