Wow, fantastic. This is super helpful. Thanks @Sam!

@Brian would you be able to provide a link to the part of the stacks project that you're referring to? I'd be very interested in an exposition of this as well. Thanks!

@DylanWilson why restrict oneself to flat extensions? I'm saying for arbitrary $R$, look at the $R$-local category, and ask for spectra which satisfy any of those criteria. There are obviously a few cases where this is known (localize at $K(n)$, or $E_n$, or $BP$).

@EricWofsey but every module over $K(n)$ is a wedge of $K(n)$'s yes?

I suspect the answer is uninteresting for harmonic spectra.

I think being able to write down the structure of cobordism rings as homotopy of Thom spectra is pretty amazing, personally.

I think it's pretty clear that I wasn't aware of this when I asked this question. Moreover, an answer would express the structure of the image of the homology of BSp in terms of one of the known bases of the homology of MSO.

So I feel bad about this, but in fact, the structure of the homology of MSO inside of the homology of MO, in terms of the Stiefel-Whitney classes, does not seem to be known (and appears to be rather hard to work out). So I'm not sure how to extend this to a full answer.

Although I'm not sure that the generators on the homology of BO that Pengelley gives are the same as your generators.

So I think the relevant reference is <a href="bit.ly/17j9KfW"> this </a>.

Thanks @Oscar! I'll try to work it out from here. I'll accept your answer once I do, and post a full solution for posterity!