Let me first say, that I am not an expert, but was interested in the same question recently, so I would also be happy if someone provides more info on the question. Let me collect some facts, which I know.
As far as I understand the general classification of orbits is in certain sense "wild" problem.
Classification up to n=7 can found in Coadjoint orbits of the group $UT(7,K)$ (2006) , further papers by A.Panov and his students give partial results on general $n$, e.g. these ones Involutions in $S_n$ and associated coadjoint orbits Diagram method in research on coadjoint orbits (2009)
There is a lots of recents studies which are "related" to the question. Especially in the case of ground field is finite. In such a case people are greatly interested in understanding representation theory of U(n,F_q) and in particular of the "orbit method" approach to it, and hence in coadjoint orbits. See some comments at mathoverflow question: Finite Unipotent Groups: References.
If you restrict to ground field to be finite, then it is worth to mention several facts: a) number of adjoint and coadjoint orbits is the same b) it is the same with the number of conjugacy classes in the group c) hence the same as number of irreps d) it is related to interesting combinatorics, see e.g. paper by A.A. Kirillov, A. Melnikov On a Remarkable Sequence of Polynomials and other papers by these and other authors.
For the finite field, these MO questions, related: Finite Unipotent Groups: References, Representation theory of p-groups in particular upper tringular matrices over F_p Irreducible representations of the unitriangular group.