The simplest case of a manifold with singularities is a manifold M with boundary, together with an identification of the boundary with a product ßM × P, where P is a fixed manifold. The associated singular space is obtained by collapsing P to a point. When P = Z/k or S1, we show how to attach to such a space a noncommutative C*-algebra that captures the extra structure. We then use this C*-algebra to give a new proof of the Freed-Melrose Z/k-index theorem and a proof of an index theorem for manifolds with S1 singularities. Our proofs apply to the real as well as to the complex case. Applications are given to the study of metrics of positive scalar curvature.