  We provide several characterisations of collapsing and noncollapsing in
convex ancient mean curvature flow, establishing in particular that collapsing
occurs if and only if the flow is asymptotic to at least one Grim hyperplane.
As a consequence, we rule out collapsing singularity models in $(n-1)$-convex
mean curvature flow (even when the initial datum is only immersed). Explicit
counterexamples show that $(n-1)$-convexity is optimal. We are also able to
rule out collapsing singularity models for suitably pinched solutions of higher
codimension.
