  This is a collection of notes on embedding problems for 3-manifolds. The main
question explored is `which 3-manifolds embed smoothly in the 4-sphere?' The
terrain of exploration is the Burton/Martelli/Matveev/Petronio census of
triangulated prime closed 3-manifolds built from 11 or less tetrahedra. There
are 13766 manifolds in the census, of which 13400 are orientable. Of the 13400
orientable manifolds, only 149 of them have hyperbolic torsion linking forms
and are thus candidates for embedability in the 4-sphere. The majority of this
paper is devoted to the embedding problem for these 149 manifolds. At present
41 are known to embed. Among the remaining manifolds, embeddings into homotopy
4-spheres are constructed for 4. 67 manifolds are known to not embed in the
4-sphere. This leaves 37 unresolved cases, of which only 3 are geometric
manifolds i.e. having a trivial JSJ-decomposition.
