(1) The Computational Homology Project offers free software CHomP that will compute homology of simplicial complexes, at least with finite field coefficients.
(2) Dionysus, from the computational topology group at Stanford, is good for computing persistent homology of Rips and Cech complexes, etc. This might be especially useful, for example, if you had points sampled from a manifold.
(3) Afra Zomorodian has apparently recently written some code for computing homology of clique (i.e. flag) complexes very quickly and with small memory requirement by going through calculations involving simplicial sets, but I don't know if the code can compute homology of arbitrary simplicial sets, and (more importantly) I don't know if the code is publicly available.