The only exposition of de Rham homology I've found is an appendix to Uranga and Ibanezs book on *String Phenomenology*. It was brief and gave only basic outline of how to construct this homology.

Now de Rhams theorem asserts that there is an isomorphism between de Rham cohomology of smooth manifolds and that of singular cohomology; and so what appears to be an invariant of smooth structure, is actually an invariant of topological structure.

Is there a similar theorem showing an isomorphism between de Rham homology and singular homology?

Differentiable manifolds. The result you want follows from Thm.16 in Sec. 21. $\endgroup$