Cohomology of $ko,tmf,MSpin,MString$ with coefficients $\mathbb{Z}/p$ for odd primes $p$

Question

It is well-known that $$H^*(ko,\mathbb{Z}/2)=\mathcal{A}\otimes_{\mathcal{A}(1)}\mathbb{Z}/2$$ $$H^*(tmf,\mathbb{Z}/2)=\mathcal{A}\otimes_{\mathcal{A}(2)}\mathbb{Z}/2$$ where $\mathcal{A}$ is the mod 2 Steenrod algebra.

$H^*(MSpin,\mathbb{Z}/2)$ and $H^*(MString,\mathbb{Z}/2)$ are closely related to the above because of the Atiyah-Bott-Shapiro orientation and Witten genus.

I find in Adams and Priddy's Uniqueness of BSO: $$H^*(ko,\mathbb{Z}/p)=\bigoplus_{s=0}^{\frac{p-3}{2}}\Sigma^{4s}\mathcal{A}_p/(\mathcal{A}_pQ_0+\mathcal{A}_pQ_1)=\bigoplus_{s=0}^{\frac{p-3}{2}}\Sigma^{4s}\mathcal{A}_p\otimes_{E(Q_0,Q_1)}\mathbb{Z}/p$$ where $\mathcal{A}_p$ is the mod $p$ Steenrod algebra for odd primes $p$ and $Q_0=\beta,Q_1=P^1\beta-\beta P^1$.

I want to know what is $H^*(MSpin,\mathbb{Z}/p)$, $H^*(tmf,\mathbb{Z}/p)$ and $H^*(MString,\mathbb{Z}/p)$ for odd primes $p$.

Any references and partial answers are appreciated.

Edit: Big thanks to Mark Behrens for his answer. As a complement, I find $$H^*(MSpin,\mathbb{Z}/p)=H^*(MSO,\mathbb{Z}/p)$$ for odd primes $p$ and the homology of $MSO$ at odd primes as a comodule over the dual Steenrod algebra is Lemma 20.38 of Switzer's textbook.

Answer 1

[Some folks started emailing me about this so I supposed I should click on the link and post what I knew...]

The homology of tmf at all primes as a comodule over the dual steenrod algebra is Theorem 21.5 of Charles Rezk's tmf notes

https://faculty.math.illinois.edu/~rezk/512-spr2001-notes.pdf

The spaces Bspin, BString are all spaces of the spectrum ko. At odd primes ko is a summand of ku, and the mod p cohomology of these spaces was determined by Bill Singer "Connective fiberings over BU and U". By the Thom isomorphism theorem, this gives an answer for MSpin and MString, but maybe not the desired answer because the isomorphism is not an isomorphism of modules over the steenrod algebra.

The more refined computation of H^*MO at odd primes was done by Rosen in his Northwestern Thesis - this computation is redone by Hovey and Ravenel in their paper "THE 7-CONNECTED COBORDISM RING AT p = 3" (despite the title it does the cohomology for all odd primes p).

Hope this helps...