This talk will cover recent joint work with Brian White. For any prescribed closed subset of a line segment in Euclidean 3-space, we construct a sequence of minimal disks that are properly embedded in a fixed open solid cylinder C around the line segment, have boundary in the boundary of C, and that have curvatures blowing up precisely at the points of the closed set. In the study of minimal varieties, the known examples of singular sets have been rather tame. We believe this result provides the first example of Cantor sets of singularities and of singular sets with non-integer Hausdorff dimension This result contrasts sharply with the global result of Colding and Minicozzi et. al: If the radius and height of the of the cylinder C both go to infinity, then the curvature-blowup set is a full line orthogonal to the extended limit lamination: a foliation of 3-space by parallel planes.