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This paper is a continuation of our work on a conjecture of Almgren onarea-minimizing surfaces with fractal singular sets. First, we prove thatarea-minimizing surfaces with fractal singular sets are prevalent on thehomology level on general manifolds. They exist even in metrics arbitrarilyclose to the flat metric on a torus. Second, we partially determine the modulispace of area-minimizing currents near the ones with fractal singular set thatwe constructed in this and previous work. Around strata of codimension at leastthree, the fractal self-intersections deform into transverse immersions undergeneric perturbations of metric. The result is sharp and we construct concreteexamples to show that our fractal singularities do not necessarily completelydissolve under generic perturbations. Finally, we improve our previous resultsto give examples with different strata limiting to each other and refine thefirst stratum to fractal subsets of finite combinatorial graphs.

Area-minimizing surfaces with fractal singular sets: prevalence, moduli space, and refinements on strata