Supposedly (I cannot find any reference, hence this question), it should hold for the standard Brownian motion $B=(B_t)_{t\ge0}$ that for almost every $\omega \in \Omega$ there exists a $t\ge 0$ such that $B(\omega)$ is 1/2-Hölder continuous at $t$.
I was trying to look that up in various sources, including ̈Mörters' and Peres' book on Brownian Motion but to no luck. Could you direct me to any source for that claim (or alternatively one that disproves it)?
Since we have almost surely $\alpha$-Hölder continuity everywhere for $\alpha<1/2$ and nowhere for $\alpha>1/2$ (as well as nowhere in any countable subset of $\mathbb{R}_+$ for $\alpha=1/2$), I assume the argument for showing the claim has to be quite fine.
A place to start would be the following two papers on Brownian "slow points": On Brownian slow points by B. Davis http://link.springer.com/article/10.1007%2FBF00532967 and On the Hausdorff dimension of the Brownian slow points by E.A. Perkins http://link.springer.com/article/10.1007%2FBF00532968, both papers date to 1983. These papers refine earlier work of J.-P. Kahane, Sur l'irrégularité locale du mouvement brownien.