Understanding how smoothly immersed, stable, minimal hypersurfaces can degenerate under uniform volume bounds is a well-known problem within geometric analysis and geometric measure theory. In low dimensions, the work of Schoen—Simon—Yau provides uniform curvature estimates, and moreover when the hypersurface is assumed to be "embedded" in a certain sense, the work of Wickramasekera provides a strong compactness and regularity theory. A key issue to understand is singular points of higher multiplicity, with flat singular points known as branch points being the main example. A priori, the topological structure about branch points could be very complicated with, for example, a sequence of "necks" degenerating toward the singularity. In this talk, I will discuss some recent results in this direction, establishing structural results near branch points where no a priori assumption on the size of the singular set is made. Some results are joint with Neshan Wickramasekera (University of Cambridge).