If you're willing for "useful" to apply to a field other than topology itself: collapses are quite important in combinatorics.  Elementary collapses correspond (via Euler characteristics) to matching up equi-numerous objects counted with opposite signs in inclusion-exclusion type problems.

Discrete Morse theory is the generalized version of this.  The basic idea in discrete Morse theory is that one can collapse faces of adjacent dimension in skeleta of a simplicial complex, then glue on higher dimensional faces in a way respecting the collapsing (without changing homotopy type).  Discrete Morse theory has been a quite important tool in topological combinatorics for the past 10 or 15 years.  See the papers of Forman: "Morse theory for cell complexes" introduced the topic (though I should mention that the basic idea was discovered by Ken Brown in "The geometry of rewriting systems: a proof of the Anick-Groves-Squier Theorem"), or "Topics in combinatorial differential topology and geometry" is a survey article.

Discrete Morse theory can also be seen as a generalization of the theory of shellings (also based on a collapsing idea), which has been important in topological and algebraic combinatorics since the late 70s/early 80s.