  In this article, we improve the partial regularity theory for minimizing
$1/2$-harmonic maps in the case where the target manifold is the
$(m-1)$-dimensional sphere. For $m\geq 3$, we show that minimizing
$1/2$-harmonic maps are smooth in dimension 2, and have a singular set of
codimension at least 3 in higher dimensions. For $m=2$, we prove that, up to an
orthogonal transformation, $x/|x|$ is the unique non trivial $0$-homogeneous
minimizing $1/2$-harmonic map from the plane into the circle $\mathbb{S}^1$. As
a corollary, each point singularity of a minimizing $1/2$-harmonic maps from a
2d domain into $\mathbb{S}^1$ has a topological charge equal to $\pm1$.
