Regarding Charles Rezk's second question:

*This leaves open the possibility that there are topological manifolds which do not admit any PL-structure but are still homeomorphic to some simplicial complex. Is this possible?*

For dimension 4, it follows from the Poincare conjecture that a 4-manifold is triangulable iff smoothable (which is also equivalent to having PL structure for dimension <8). See Problem 3 of http://www.maths.ed.ac.uk/~aar/haupt/sandro.pdf. Also see [this presentation][1].

For dimension >4, [Springer Online Reference Works][2] claims that "the imbedding $PL \subset TRI$ is also irreversible in the same strong sense (there exist polyhedral manifolds of dimension $\geq 5$ that are homotopy inequivalent to any PL-manifold)",  but gives no examples. In [this presentation][3] it is stated that "All oriented closed 5-manifolds triangulable", so I think among them there may be some with nontrivial KS invariant and hence cannot bear PL structure.

In addition, [This book][4] (p.168, Theorem 18.4) seems to conatain a result that strengthens the one mentioned in Paul's answer.

Added: [This paper][5] (22.5. Example) explicitly gives an example of "A topological manifold which is homeomorphic to a polyhedron but does not admit any PL structure".


  [1]: http://math.uci.edu/~rstern/Hiroshima2_18_06.pdf
  [2]: http://eom.springer.de/t/t093230.htm
  [3]: http://math.uci.edu/~rstern/Hiroshima2_18_06.pdf
  [4]: http://books.google.com/books?id=gz2CbgR5RbwC&printsec=frontcover&source=gbs_atb#v=onepage&q&f=false
  [5]: http://arxiv.org/abs/math.AT/0105047