This is answering a slightly different question, but here goes:
If the question is "does knowing the singular (co)chains up to quasi-isomorphism determine the space up to weak equivalence?" then of course the answer is no. To some extent the answer can be turned into a yes by altering the question, considering the (co)multiplication on (co)chains as part of the structure that a quasi-isomorphism must preserve. Mike Mandell has made this idea precise (using operads) and proved powerful theorems along these lines, but I'm not an expert.
Of course, the operad structure on chains of course determines the cup product, the Steenrod operations, and more (e.g. differentials in Adams spectral sequence). EDIT: And Massey products.
