With J. Nagy and S. Seuret we have studied multifractal properties of the Hölder spectrum of several different typical/generic (in the sense of Baire) objects: monotone functions, measures, continuous monotone increasing in several variables (MISV) functions. In this talk I will focus on the two most recent developments concerning multifractal Hölder spectrum of convex hulls of typical continuous functions and of typical convex functions defined on the d-dimensional closed unit cube.