The conjecture in question can also be thought of as the $4$-dimensional PL Poincare conjecture (because low-dimensional PL manifolds, including those of dimension $4$, carry a unique smooth structure) and this is how it is understood in most references mentioned below.
Some interesting approaches to the conjecture and its special cases can be found in several papers by Frank Quinn (some based on TQFTs and others in more classical spirit) and in some papers by Robert Craggs.
Much of the effort has been focused on the group-theoretic Andrews-Curtis conjecture, whose validity would imply that PL (or smooth) homotopy $4$-spheres given as handlebodies without $3$-handles are PL (or smoothly) standard. The latter assertion would also follow from the "Generalized Property R" conjecture. Then there's a separate industry of finding handlebody presentations of simply-connected $4$-manifolds without $3$-handles (see Problems 4.18 and 4.73 in Kirby's list, Section 6 here, Gadgil's preprint and Quinn's Corollary 3.2; note also Rasmussen's withdrawn paper arxiv.org/abs/1005.4674).
As observed by Curtis in an earlier paper (in "Topology of 3-manifolds and related topics"), every compact contractible $2$-polyhedron PL embeds in some PL homotopy $4$-sphere; so if you find a compact $2$-polyhedron one that doesn't PL embed in $S^4$, you're done with the 4D PL Poincare conjecture. This line of attack inspired some literature on PL embeddings of acyclic $2$-polyhedra in $S^4$ starting I guess with Zeeman's dunce hat paper; see this review for additional references.
There are numerous other approaches and related techniques, e.g. "Gluck twists" and "Akbulut corks". Kirby's problem list is a good source of further references prior to mid-90s; some other basic references on the Andrews-Curtis conjecture are collected here under (O1).