Using results of Colding-Minicozzi and an extension due to Meeks, we prove that a sequence of properly embedded minimal disks in a 3-ball must have a subsequence whose curvature blow-up set lies in a union of disjoint $C^1$curves.
"Curvatures of embedded minimal disks blow up on subsets of $C^1$ curves." J. Differential Geom. 100 (2) 389 - 394, June 2015. https://doi.org/10.4310/jdg/1430744125