  Cosmological $\alpha$-attractors stand out as particularly compelling models
to describe inflation in the very early universe, naturally meeting tight
observational bounds from cosmic microwave background (CMB) experiments. We
investigate $\alpha$-attractor potentials in the presence of an inflection
point, leading to enhanced curvature perturbations on small scales. We study
both single- and multi-field models, driven by scalar fields living on a
hyperbolic field space. In the single-field case, ultra-slow-roll dynamics at
the inflection point is responsible for the growth of the power spectrum, while
in the multi-field set-up we study the effect of geometrical destabilisation
and non geodesic motion in field space. The two mechanisms can in principle be
distinguished through the spectral shape of the resulting scalar power spectrum
on small scales. These enhanced scalar perturbations can lead to primordial
black hole (PBH) production and second-order gravitational wave (GW)
generation. Due to the existence of universal predictions in
$\alpha$-attractors, consistency with current CMB constraints on the
large-scale spectral tilt implies that PBHs can only be produced with masses
smaller than $10^8\,\text{g}$ and are accompanied by ultra-high frequency GWs,
with a peak expected to be at frequencies of order $10\,\text{kHz}$ or above.
