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 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.