A paper on the arXiv by Sergiy Maksymenko  caught my eye the other day. (It was posted about a year ago, but I saw it when it was updated.) The paper considers the following problem: Let f be a Morse function on a surface S and let F be the orbit of f in under all diffeomorphisms of S. (The orbit is a subset of the space of all smooth functions on S.) You can put a topology called the Whitney topology on , which induces a topology on the orbit set. Maksymenko shows that this space is homotopy equivalent to a finite cell complex and that the fundamental group of this subcomplex is a subgroup of some braid group over that surface.
There are two reasons this caught my eye. First, I like this idea of considering the space of topological objects that are isotopic to a fixed subset or structure on a larger manifold. For example, Hatcher  has looked at the space of knots isotopic to a fixed knot in the 3-sphere. The homotopy type of such a space is determined by a combination of the homotopy type of the diffeomorphism group of the original manifold and by the symmetries of the subset or structure.
The second reason is that it reminds me of Hatcher and Thurston’s paper on mapping class groups of surfaces , in which they define the pants complex of a surface. Rather than finding a cell complex that encodes a single orbit inside , they find a cell complex that encodes the overall structure of , with each vertex representing a collection of orbits (all of which induce the same pants decomposition) and with edges and higher dimensional cell encoding how the orbits fit together to form the entire space. The idea of approximating the topology of an infinite dimensional space with a cell complex comes up less explicitly in other areas as well, such as thin position and even the stabilization problem for Heegaard splitings.
Before I end this post, I just wanted to suggest one more thing: One could ask this same question for Morse functions in any smooth manifold. For an irreducible 3-manifold, I would guess that the space is always contractible, but that’s just a guess, not a conjecture. For a reducible 3-manifold and a Morse function in which a level surface is a reducing sphere, it seems like there’s room for a non-trivial loop in the orbit space, though maybe not. But this seems like an interesting problem.
 Hatcher, A.; Thurston, W. A presentation for the mapping class group of a closed orientable surface. Topology 19 (1980), no. 3, 221–237.