Recall that a (graded) module $V^\ast$ over the Steenrod algebra $\mathcal A^\ast$ is said to be unstable if $Sq^i v = 0$ for $i > |v|$. The motivating example, of course, is that if $V^\ast = H^\ast(X)$ for a space $X$ with its natural $\mathcal A^\ast$ structure, then $V^\ast$ is unstable.

The category of $\mathcal A^\ast$-modules which are finite-dimensional over $\mathbb F_p$ is dual to the category of $\mathcal A_\ast$-comodules which are finite-dimensional over $\mathbb F_p$, where $\mathcal A_\ast$ is the dual Steenrod algebra. So the instability condition should be expressible from this dual perspective.

Question 1: Let $V_\ast$ be a finite-dimensional graded $\mathbb F_p$-vector space equipped with the structure of an $\mathcal A_\ast$-comodule. Under what conditions is the dual vector space $V^\ast$ unstable (with its natural $\mathcal A^\ast$-module structure)?

Ideally, the condition would be expressed in terms of Milnor's $\{\xi_m, \tau_n\}$ generators. In principle it should be straightforward to do the translation, but I am a bit intimidated by the Adem relations. I am also stuck already because even before dualizing, I don't know how to express the instability condition in terms of the generating set $\{Sq^{2^k} Sq^{2^{k-1}}\cdots Sq^1\}$, which I suppose leads to a subsidiary question:

Question 2: Let $V^\ast$ be a finite-dimensional $\mathcal A^\ast$-module. In terms of the generating set $\{Sq^{2^k} Sq^{2^{k-1}} \cdots Sq^1\}$, when is $V^\ast$ unstable?

I'd be happy to know the answer for $p=2$, $p>2$, or both.