examples of k-invariants of spectra

Question

The homotopy groups of commonly used topological spectra (like KO, S, MO, MSO, etc) are easy to find in literature, even appearing on Wikipedia's List of Cohomology Theories; however, I have had some difficulty finding descriptions of these spectra's k-invariants (aka Postnikov invariants). Does anyone have a reference detailing the k-invariants of common spectra? I'm particularly interested in the spectra noted above as well as MSpin and so-called Supercohomology ($\pi_0=\mathbb{Z}$, $\pi_2=\mathbb{Z}_2$).