  There is a well-known stratification of the moduli space $M_g$ of
Deligne-Mumford stable curves of genus $g$ by the loci of stable curves with a
fixed number $i$ of nodes, where $i \le 3g-3$. The associated moduli stack
${\cal M}_g$ admits an analogous stratification.
  Our main objects of study are those one-dimensional substacks of the moduli
stack, which are the irreducible components of the stratum corresponding to
stable curves with exactly $3g-4$ nodes. We describe these substacks
explicitely as quotient stacks, and relate them to other, and simpler, moduli
stacks of (permutation classes of) pointed stable curves. In an appendix, an
extensive compendium on quotient stacks is provided.
