It is classical that the zero set and the set of record times of a linear Brownian motion have Hausdorff dimension 1/2 almost surely. Can we find a larger random set on which a Brownian motion is monotone? Perhaps surprisingly, the answer is negative. We outline the short proof, which is an application of Kaufman's dimension doubling theorem for planar Brownian motion. If time permits, we discuss related results for random walk and fractional Brownian motion as well, and pose some open problems. This is a joint work with Omer Angel, András Máthé, and Yuval Peres.