  In this paper we study the singular set of Dirichlet-minimizing $Q$-valued
maps from $\mathbb{R}^m$ into a smooth compact manifold $\mathcal{N}$ without
boundary. Similarly to what happens in the case of single valued minimizing
harmonic maps, we show that this set is always $(m-3)$-rectifiable with uniform
Minkowski bounds. Moreover, as opposed to the single valued case, we prove that
the target $\mathcal{N}$ being non-positively curved but not simply connected
does not imply continuity of the map.
