We show that almost every function (in the sense of prevalence) in a Sobolev space is multifractal: Its regularity changes from point to point; the sets of points with a given Hölder regularity are fractal sets, and we determine their Hausdorff dimension.
"How smooth is almost every function in a Sobolev space?." Rev. Mat. Iberoamericana 22 (2) 663 - 682, September, 2006.