  We study fine structural properties related to the interior regularity of
$m$-dimensional area minimizing currents mod$(q)$ in arbitrary codimension. We
show: (i) the set of points where at least one tangent cone is translation
invariant along $m-1$ directions is locally a connected $C^{1,\beta}$
submanifold, and moreover such points have unique tangent cones; (ii) the
remaining part of the singular set is countably $(m-2)$-rectifiable, with a
unique flat tangent cone (with multiplicity) at $\mathcal{H}^{m-2}$-a.e. flat
singular point. These results are consequences of fine excess decay theorems as
well as almost monotonicity of a universal frequency function.
