A compact subset $K$ of the complex plane is said to be $W^{1,p}$ removable if any continuous real-valued function $f$ on $\mathbb{C}$ which is in the Sobolev space $W^{1,p}(\mathbb{C} \setminus K)$ is automatically also in the Sobolev space $W^{1,p}(\mathbb{C})$. This property is true for all values of $p$ for points and line segments, but false for sets with non-empty interior, and in general there is no simple condition to determine whether a given set is removable or not. In certain cases, there is a link between removability and how “rough” the set $K$ is. In particular, if $K$ is the graph of a function, it is known that its Hölder continuity is related to its removability. We will present new results on the removability of the graph of a one-dimensional Brownian motion on an interval and show that it is almost surely not $W^{1,p}$ removable for finite $p$, but is removable for $p = \infty$. This talk is based on joint work with Jason Miller.