  The well-rounded retract for $\mathrm{SL}_n(\mathbb{Z})$ is defined as the
set of flat tori of unit volume and dimension $n$ whose systoles generate a
finite-index subgroup in homology. This set forms an equivariant spine of
minimal dimension for the space of flat tori.
  For both the Outer space $X_n$ of metric graphs of rank $n$ and the
Teichm\"uller space $\mathcal{T}_g$ of closed hyperbolic surfaces of genus $g$,
we show that the literal analogue of the well-rounded retract does not contain
an equivariant spine. We also prove that the sets of graphs whose systoles fill
either topologically or geometrically (two analogues of a set proposed as a
spine for $\mathcal{T}_g$ by Thurston) are spines for $X_n$ but that their
dimension is larger than the virtual cohomological dimension of
$\mathrm{Out}(F_n)$ in general.
