  A triangulation of a $3$-manifold can be shown to be homeomorphic to the
$3$-sphere by describing a discrete Morse function on it with only two critical
faces, that is, a sequence of elementary collapses from the triangulation with
one tetrahedron removed down to a single vertex. Unfortunately, deciding
whether such a sequence exist is believed to be very difficult in general.
  In this article we present a method, based on uniform spanning trees, to
estimate how difficult it is to collapse a given $3$-sphere triangulation after
removing a tetrahedron. In addition we show that out of all $3$-sphere
triangulations with eight vertices or less, exactly $22$ admit a non-collapsing
sequence onto a contractible non-collapsible $2$-complex. As a side product we
classify all minimal triangulations of the dunce hat, and all contractible
non-collapsible $2$-complexes with at most $18$ triangles. This is complemented
by large scale experiments on the collapsing difficulty of $9$- and $10$-vertex
spheres.
  Finally, we propose an easy-to-compute characterisation of $3$-sphere
triangulations which experimentally exhibit a low proportion of collapsing
sequences, leading to a heuristic to produce $3$-sphere triangulations with
difficult combinatorial properties.
