I will outline the problem of determining which 3-manifolds embed in the 4-sphere. This will have essentially 3 parts:
(i) Known obstructions to a 3-manifold embedding in the 4-sphere such as torsion linking forms, Alexander polynomials, the Rochlin invariant, the Oszvath-Szabo d-invariant, Casson-Gordon invariants.
(ii) Procedures for constructing embeddings.
(iii) Computations: Recent independent work of Matveev, Petronio and Burton has produced lists of all the 3-manifolds (up to homeomorphism) that can be triangulated with 11 or less tetrahedra.
Computer implementation of known obstructions gives gives us a list of all 3-manifolds which pass the tests (1) but for which we still, as of yet, do not know if they embed in S^4. I will describe some characteristic manifolds from this list.