  We consider two established supergravities which are known to provide the
low-energy effective description of either superstring theory or M-theory: the
six-dimensional theory of Romans, and the maximal supergravity in seven
dimensions. We compactify on an $S^{1}$ and $T^{2}$, respectively, to obtain 5D
sigma-models coupled to gravity, and spectra of bosonic excitations are
computed numerically by fluctuating fields on backgrounds which holographically
realise confinement. We propose a method to detect mixing effects between
scalar resonances and the pseudo-Nambu--Goldstone boson associated with
spontaneous breaking of conformal invariance: the dilaton. This test consists
of neglecting a key component of the spin-0 fluctuation variables, effectively
disregarding their back-reaction on the underlying geometry; where
discrepancies arise compared to the proper calculation we infer dilaton mixing.
In both cases this analysis evinces a parametrically light dilaton. For each
theory we also uncover a tachyonic instability within their parameter space. We
hence proceed to investigate their respective phase structures, reasoning that
some mechanism must render the instabilities physically inaccessible. We
compile a catalogue of geometrically distinct backgrounds within each theory,
and derive general expressions for their holographically renormalised free
energy $F$. Another numerical routine is used to systematically extract data
for some special deformation parameters, and $F$ is plotted in units of an
appropriate universal scale. Our analysis proves fruitful: each theory shows
evidence of a first-order phase transition which induces the spontaneous
decompactification of the shrinking dimension before the instability manifests,
favouring instead a branch of singular solutions. The light dilaton resonance
appears only along a metastable portion of the branch of confining backgrounds.
