Let S be a surface of genus g with some parked points (n of them). Assume $n \geq 2$ and fix two of the marked points. Consider the set of embedded arcs going between these two special points. The group of diffeomorphisms preserving the marked points acts on this set. What are the orbits? How many are there? Equivalently, we can instead consider embedded arcs up to isotopy and consider the action of the mapping class group of the surface.
I am specifically interested in two cases: genus zero with four marked points, and genus 1 with two marked points. However I think the general question is also interesting and it seems like the sort of thing that has been studied before. I just no idea where to look.