The intermittent transition between slow growth and rapid shrinkage in polymeric assemblies is termed dynamic instability, a feature observed in a variety of biochemically distinct assemblies including microtubules, actin and their bacterial analogs. The existence of this labile phase of a polymer has many functional consequences in cytoskeletal dynamics, and its repeated appearance suggests that it is relatively easy to evolve. Here, we consider the minimal ingredients for the existence of dynamic instability by considering a single polymorphic filament that grows by binding to a substrate, undergoes a conformation change, and may unbind as a consequence of the residual strains induced by this change. We identify two parameters that control the phase space of possibilities for the filament: a structural mechanical parameter that characterizes the ratio of the bond strengths along the filament to those with the substrate (or equivalently the ratio of longitudinal to lateral interactions in an assembly), and a kinetic parameter that characterizes the ratio of time scales for growth and conformation change. In the deterministic limit, these parameters serve to demarcate a region of uninterrupted growth from that of collapse. However, in the presence of disorder in either the structural or the kinetic parameter the growth and collapse phases can coexist where the filament can grow slowly, shrink rapidly, and transition between these phases, thus exhibiting dynamic instability. We exhibit the window for the existence of dynamic instability in a phase diagram that allows us to quantify the evolvability of this labile phase.