Yes, the manifold question is completely answered [here][1]. Namely $\exp_n X^k$ (where $X^k$ is a $k$-manifold) is a manifold *if and only if* $k=1, n=3$ or $k=n=2.$ This is Theorem 1.3 in the referenced paper. **EDIT** Also, of course if $n=1,$ though the authors overlook this...


  [1]: http://arxiv.org/pdf/0902.3773v2

**EDIT** By the way, some very nice papers on the subject have been written by Chris Tuffley (a couple seem to be in AGT).

**ANOTHER EDIT** In particular, Tuffley gives the simplicial complex structure of $\exp_n S^1$ explicitly, and also describes the "complement" of $\exp_{n-2}S^1$ in $\exp_n S^1,$ which is already interesting in the case $n=3$ (it's the trefoil knot complement). From Tuffley's thing you can, at least in principle, answer all homeomorpism-related questions (I am referring to the paper with the illogical title: "Finite Subset Spaces of $S^1.$"