  Energy-minimizing constraint maps are a natural extension of the obstacle
problem within a vectorial framework. Due to inherent topological constraints,
these maps manifest a diverse structure that includes singularities similar to
harmonic maps, branch points reminiscent of minimal surfaces, and the intricate
free-boundary behavior of the obstacle problem. The complexity of these maps
poses significant challenges to their analysis.
  In this paper, we first focus on constraint maps with uniformly convex
obstacles and establish continuity (and therefore higher-order regularity)
within a uniform neighborhood of the free boundary. More precisely, thanks to a
new quantitative unique continuation principle near singularities (which is new
even in the setting of classical harmonic maps), we prove that, in the
uniformly convex setting, topological singularities can only lie in the
interior of the contact set. We then establish the optimality of this result.
  Second, while exploring the structure of the free boundary, we investigate
the presence of branch points and show how they lead to completely new types of
singularities not present in the scalar case.
