  We construct a smooth Deligne-Mumford compactification for the moduli space
of curves with an m-tuple of spin structures using line bundles on quasi-stable
curves as limiting objects, as opposed to line bundles on stacky curves. For
all m, we give a combinatorial description of the local structure of the
corresponding coarse moduli spaces. We also classify all irreducible and
connected components of the resulting moduli spaces of multiple-spin curves.
