  We study the limiting behavior of minimizing $p$-harmonic maps from a bounded
Lipschitz domain $\Omega \subset \mathbb{R}^{3}$ to a compact connected
Riemannian manifold without boundary and with finite fundamental group as $p
\nearrow 2$. We prove that there exists a closed set $S_{*}$ of finite length
such that minimizing $p$-harmonic maps converge to a locally minimizing
harmonic map in $\Omega \setminus S_{*}$. We prove that locally inside $\Omega$
the singular set $S_{*}$ is a finite union of straight line segments, and it
minimizes the mass in the appropriate class of admissible chains. Furthermore,
we establish local and global estimates for the limiting singular harmonic map.
Under additional assumptions, we prove that globally in $\overline{\Omega}$ the
set $S_{*}$ is a finite union of straight line segments, and it minimizes the
mass in the appropriate class of admissible chains, which is defined by a given
boundary datum and $\Omega$.
