I was just looking through a book which proves many interesting and rather difficult results on Brownian motion (<a href="http://people.bath.ac.uk/maspm/book.pdf">pdf link</a>, <a href="http://people.bath.ac.uk/maspm">website link</a>), and it seems that the Kolmogorov zero-one law applies to most of these.

Using Fourier transforms, a standard Brownian motion X<sub>t</sub> on the range 0&le;t&le;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, B<sub>n</sub>, C<sub>n</sub> 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).

It gets more interesting if you look at the modulus of continuity of Brownian motion. For any time t, the Law of the iterated logarithm says that
$$
\limsup_{h\to 0}\frac{|X_{t+h}-X_t|}{\sqrt{2h\log\log (1/h)}}=1
$$
with probability 1. From this, you can say that, with probability one, Brownian motion satisfies this limit almost everywhere (but not everywhere - there are exceptional times).
More generally, they show that with probability one, the following are true at all times,
$$
\limsup_{h\to 0}\frac{|X_{t+h}-X_t|}{\sqrt{2h\log(1/h)}}\le 1,\ 
\limsup_{h\to 0}\frac{|X_{t+h}-X_t|}{\sqrt{h}}\ge 1.
$$
With probability one, these bounds are achieved. Times where the left inequality is an equality are <em>fast times</em> and <em>slow times</em> are when the right one is an equality. In the book I linked, they calculate lots of stuff about these fast and slow times, such as their fractal dimensions.

According to my decomposition of Brownian motion above, all of these definitions and statements are about tail events and we know that they must have probability 0 or 1 of being true. In fact, the sets of slow and fast times are defined in terms of tail events, so <em>any</em> measurable statement about these sets must be either always true or always false with probability one, and any measurable function of them, such as their fractal dimensions, must be deterministic constants with probability one, even though it is hard to calculate what they are. The same goes for many of the other properties of Brownian motion in the book I linked - they are tail events and therefore always true or false with probability one.