Long-wavelength models for van der Waals driven rupture of a free thin viscous sheet and for capillary pinchoff of a viscous fluid thread both give rise to families of first-type similarity solutions. The scaling exponents in these solutions are independent of the dimensionality of problem. However, the structure of the similarity solutions exhibits an intriguing geometric dependence on the dimensionality of the system: van der Waals driven sheet rupture proceeds symmetrically, whereas thread rupture is inherently asymmetric. To study the bifurcation of rupture from symmetric to asymmetric forms, we generalize the governing equations with the dimension serving as a control parameter. The bifurcation is governed by leading-order inviscid dynamics in which viscous effects are asymptotically small but nevertheless provide the selection mechanism.