Many process systems, such as distillation columns and other equipment with phase change, exhibit multiple modes of physical behavior that can be described by non-differentiable (i.e., nonsmooth) models. In this paper, we introduce a nonsmooth model for steady-state multistage distillation that can describe columns with dry and/or vaporless stages reliably. The model consists of a system of nonsmooth MESH and specification equations, without inequality or complementarity constraints, that can be directly solved with the semismooth Newton method using automatically computed generalized derivatives. With a modified version of pseudo-arclength continuation, we have been able to observe several novel types of bifurcations in dry and/or vaporless distillation columns. Many of the bifurcations exhibit degenerate behavior with an infinite number of steady states for certain critical input specifications, and occur in general multistage distillation systems regardless of the mixture components or thermodynamic models chosen. We present case studies drawn from the literature and analyze the occurrence and behavior of the bifurcations with respect to several types of column configurations, involving ideal stages, stage efficiencies, pressure gradients, tray heat transfer, multiple feeds, and side products. The associated bifurcation curves are inherently nonsmooth and can be described mathematically by the concept of piecewise-smooth manifolds introduced in this paper.