There have been a couple of recent results which demonstrate that the class of subgroups of one-relator groups is very rich.  For instance, Calegari--Walker proved that a random 1-relator group contains a surface subgroup, and Calegari and I improved this by demonstrating that a random 1-relator group contains a subgroup isomorphic to the fundamental group of an acylindrical hyperbolic 3-manifold.  Both papers are on the arXiv.

A random 1-relator group is hyperbolic (since its relator satisfies the C'(1/6) small-cancellation condition) and 2-dimensional.  Results of Bowditch and Kapovich--Kleiner then impose certain restrictions on the (quasiconvex) subgroups that can arise---see the answer to [this][1] MO question for some details.  In a sense, the above results show that the quasiconvex subgroups of random 1-relator groups are as rich as possible.  This is discussed in the introduction to our paper.

This leaves two strands uncovered.  A random 1-relator group will also contain non-quasiconvex finitely generated subgroups. (I think Dunfield--Thurston proved this in the 2-generator case.)  And, of course, plenty of 1-relator groups do not behave like a `random' one---for instance, Baumslag's famous example of a no-abelian 1-relator group with every finite quotient cyclic.


  [1]: http://mathoverflow.net/questions/135788/topology-of-boundaries-of-hyperbolic-groups/135794#135794