We study the geometric behavior of the neutral modes of the linearized equations from four directional-solidification models in terms of singularity theory. These equations admit well-known diffusive or Mullins-Sekerka instabilities. For a range of solidification velocity ${\mathit{V}}_{\mathit{c}}$V${\mathit{V}}_{\mathit{a}}$, a planar solidification front is linearly unstable to a range of disturbances. The standard neutral curve in linear theory exhibits weak wavelength selection for V near ${\mathit{V}}_{\mathit{c}}$. An equivalence transformation of the neutral-stability relation of the nonsymmetric model distinguishes the basic geometric behavior of this system. The neutral-stability relation is an unfolding of a cubic cuspoid normal form. The solution space of this system is a cusp manifold, generated by families of neutral curves each forming a path in the manifold. This manifold connects two unfolding theories, allowing us to show the sense in which ${\mathit{V}}_{\mathit{c}}$ and ${\mathit{V}}_{\mathit{a}}$ parametrize degenerate singular points and to show that these points are structurally unstable as critical points. We show that wavelength selection is enhanced in the neutral curves of the transformed system, and that the singularity set of the manifold demarcates stability regions solely in terms of control variables. A particular neutral curve will be open or closed depending on how its path crosses the singularity set, and the system will not admit hysteresis. The weak wavelength selection is due not only to the thermophysical or control parameters of the material, but to the singular behavior that is intrinsic to the formulation. We show that four solidification models possess cubic normal forms, and that asymptotic limits reveal two different normal forms. The main goal is to point out the geometric structure of a model, and to show how one can distinguish different solidification formulations and asymptotic limits, using simple geometric criteria.