There are two main results in this paper. The first gives a description of the homotopy types of the spectra \(k<n>\) corresponding to bordism theories with singularities for which \(\pi_*(k<n>)=({\mathbb{Z}}/p)[t]\) with dim t\(=2(p^ n-1)\) for a prime p. The main tool used is the Postnikov tower of such a spectrum \(k<n>\), together with the corresponding invariants which are higher-order cohomology operations denoted \(\tilde Q_ n^{(s)}\). The second result gives a necessary and sufficient condition for a vector bundle or sphere bundle to be \(k<n>\)-orientable, namely all the operations \(\tilde Q_ n^{(s)}\) must act trivially on the Thom class of the bundle.