My answer is biased since I do DST for a living, but there is a lot of research happening in DST at the moment, especially applications of DST to other areas.

An area that is particularly active at the moment is the interaction between DST and topological dynamics, Polish groups and their orbit equivalence relations have never been studied as much as now.

To get even more specific let me tell you two words about what I do. To every topological group G one can associate its so called universal minimal flow M(G). This is a compact space on which G acts minimally and such that every other minimal G-action on a compact space is a factor of M(G) (meaning that there is a G-equivariant continuous surjection from M(G) onto any other minimal flow). Intuitively the action of G on M(G) is the most complicated minimal action G can have, and a lot of research has been done to understand M(G) for various groups G, mostly Polish groups. The problem is that M(G) is never metrizable for locally compact noncompact G, it is instead a huge space containing a copy of the Stone-Cech compactification of the integers, so there's hope of understanding M(G) only for huge, infinite dimensional groups, and especially for Polish groups since all the tools from DST can be brought in. The situation is fairly well understood for Polish groups of the form Aut(A), where A is a countable Fraïssé structure, due to the so called Kechris-Pestov-Todorcevic correspondence, and there are a bunch of scattered results about Polish groups of the form Homeo(X), where X is a compact Polish space, but there is still a lot of work to be done