We investigated the dynamics of layering transitions and other structure formation processes in molecularly thin liquid films upon reducing the separation between two atomically smooth mica substrates. Using a newly developed surface forces apparatus with two-dimensional imaging capability, we followed the hydrodynamic processes during drainage with unprecedented precision. Depending on the substrate elasticity and the approach rate, drainage occurs either in a series of consecutive layering transitions or in a single step. In the latter case, nanoscopic amounts of liquid are trapped inside the contact area transiently. The experimental observations are explained qualitatively by combining hydrodynamic effects with elastic deformations of the substrates. Furthermore, we present evidence for anisotropy in the fluid dynamics induced by the lattice symmetry of the substrates.