I was just looking through a book which proves many interesting and rather difficult results on Brownian motion (pdf link, website link), and it seems that the Kolmogorov zero-one law applies to most of these.
Using Fourier transforms, a standard Brownian motion Xt on the range 0≤t≤1 can be decomposed as $$ X_t = At + \sum_{n=1}^\infty\frac{1}{\sqrt{2}\pi n}\left(B_n(\cos 2\pi nt - 1)+C_n\sin 2\pi nt\right) $$ where A, Bn, Cn are independent normals with mean 0 and variance 1. It follows that any property of the Brownian motion which is unchanged under addition of a linear combination of sines, cosines and linear terms is a tail event and, by Kolmogorov's zero-one law, has probability zero or one. Eg, Brownian motion is known to be nowhere differentiable (with probability 1).