In general, these are incredibly hard questions. It seems to me that one natural question to ask (if you are interested in $\pi_0$ of ring spectra) would be about understanding even periodic $\mathbf{E}_\infty$-rings $A$ with $\pi_0 A = R$. Alternatively, you could attempt to understand those $\mathbf{E}_\infty$-rings $A$ with $\pi_\ast A = R_\ast$, where $R_\ast$ is some graded ring. In what follows, I'll attempt to address some of these questions (in particular, this is not a complete answer to your question). Let me begin with some generalities.

If you restrict to asking for $K(1)$-local $\mathbf{E}_\infty$-rings $A$ with $\pi_0 A = R$, then there are a few things you can say. For instance, if $p=0$ in $R$, then the collection of such $A$ is necessarily empty if $R$ is nonzero. If $R$ does not have a lift of Frobenius, then the collection of such $A$ is again empty. (Examples of such non-liftability results, in the $K(d)$-local $\mathbf{E}_n$-case for varying $d$ and $n$, are in the paper referred to above; there's an updated version on my webpage, which hasn't yet found its way onto the arXiv.)

There are also some positive results: suppose $R_\ast$ is an even-periodic graded $p$-torsion free ring such that $R_0$ is the $p$-adic completion of a smooth $\mathbf{Z}_p$-algebra, which is equipped with:

• a formal group classified by a map $MUP_0 \to R_0$ for which the induced formal group over $R_0/p$ is of height $1$, and
• an action of $\mathbf{Z}_p^\times$ and a compatible $p$-derivation (a "$\theta$-algebra structure").

Then there is a(n even periodic, complex oriented) $K(1)$-local $\mathbf{E}_\infty$-ring $A$ such that $\pi_\ast A = R_\ast$. This can be deduced from the results in the following paper of Lawson's and Naumann's: https://arxiv.org/abs/1101.3897. Admittedly, the conditions specified above seem like a lot; however, I know of no way to get past this. I also do not know how one might generalize this to constructing $K(n)$-local $\mathbf{E}_\infty$-rings for $n\geq 2$. The obstruction stems from the fact that we just do not know as much about power operations at heights $\geq 2$ as we do at height $1$. (This is not to downplay the work of Rezk, Zhu, and others on height $2$ power operations.)

Let me now give some examples. Fix an $\mathbf{E}_\infty$-ring $A$.

• Suppose $\pi_0 A = \mathbf{Q}$. Then $A$ is an $\mathbf{E}_\infty$-$\mathbf{Q}$-algebra, and these are very well understood. Indeed, $\mathbf{E}_\infty$-$\mathbf{Q}$-algebras are the same as commutative dg-$\mathbf{Q}$-algebras. The same is true of any $\mathbf{E}_\infty$-$R$-algebras over any (ordinary) $\mathbf{Q}$-algebra $R$ (hence, in particular, $\mathbf{Q}_p$ and $\overline{\mathbf{Q}}$).

• Suppose $\pi_0 A = \mathbf{F}_p$. If $A$ is $p$-local, then a result of Hopkins and Mahowald implies that $A$ is an $\mathbf{E}_\infty$-$\mathbf{F}_p$-algebra. There can be many such $A$, because Steenrod operations exist. For example, take the $\mathbf{F}_2$-algebras $A_1 = \mathbf{F}_p \otimes^\mathbf{L}_{\mathbf{Z}} \mathbf{F}_p$ (the fiber product in "derived schemes") and $A_2 = \mathbf{F}_p \otimes_{\mathbb{S}} \mathbf{F}_p$ (the fiber product in "spectral schemes"). Then $\pi_\ast A_1 = \mathbf{F}_p[t]/t^2$ with $|t|=1$ (so $\pi_0 A_1 = \mathbf{F}_p$), but $\pi_\ast A_2$ is the mod $p$ dual Steenrod algebra (so $\pi_0 A_2 = \mathbf{F}_p$ as well), but one is clearly much larger than the other! This gives an example showing that (in your notation) $\Sigma^\infty(\mathbf{F}_p) \supsetneq \Sigma^\triangledown(\mathbf{F}_p)$.

• Suppose $\pi_0 A = \mathbf{Z}$. Then, there are a ton of possibilities for $A$. For instance, $A$ could be $\mathbb{S}, MU, KU, KO, TMF, Tmf, ...$ (and their connective covers). Again, even periodicity would be a reasonable condition to impose. Then, the number of examples is cut down significantly (in the list provided above, for instance, you're only left with $KU$).

Hopefully someone else more knowledgeable in these topics can provide a more comprehensive and satisfactory answer to your question.