This work is divided into two, largely independent parts. The first discusses so-called two-valued minimals, and takes up the majority of the text. It concludes with a proof of a Bernstein-type theorem valid in dimension four: in this dimension entire two-valued minimal graphs are linear. In gross terms we follow a strategy similar to that used to prove the Bernstein theorem for single-valued graphs; for example we prove interior gradient and area estimates which echo those available in the classical theory. The main contrast with these historical results is the possible presence of a large set of singularities. This is exacerbated by the fact that two-valued minimal graphs do not minimise area, unlike their single-valued counterparts. As a consequence the space of surfaces which could arise as weak limits is potentially huge. This includes the so-called tangent and blowdown cones, which respectively approximate the infinitesimal behaviour near singular points and the asymptotic behaviour at large scales. Of special interest are a subclass we call classical cones, as they provide local models near particularly large sets of singularities. The classification of these, which we establish in dimensions up to seven, represents one of the main technical challenges of our work. In dimension four, we are able to push this further and give a proof of the aforementioned Bernstein-type theorem.

The second part deals with minimal arising from a semilinear elliptic PDE called the Allen-Cahn equation. There we prove a spectral lower bound for hypersurfaces that arise from sequences of critical points with bounded indices. In particular, the index of two-sided minimal hypersurfaces constructed using multi-parameter Allen-Cahn min-max methods is bounded above by the number of parameters used in the construction. Finally, we point out by an elementary inductive argument how the regularity of the hypersurface follows from the corresponding result in the stable case.