As the title says, I'm being asked to characterize all $\mathbb{k}[X]$-modules of dimension $1,2$ and $3$ for the given fields.
Since a module $V$ on $\mathbb[X]$ is a $\mathbb{k}$-vector space with a multiplication given by $pm = T(p)(m)$ for some $T \in \operatorname{End}_{\mathbb{k}}(V)$, the $1$-dimensional case seems pretty straighforward: it must be that $T \simeq (m \mapsto \lambda m)$ for some $\lambda \in \mathbb{k}$, and so one can see that $V \simeq \mathbb{k}$ with the multiplication given by $pv = p(\lambda)v$ which I will denote $\mathbb{k}_\lambda$.
However, for the $2$ and $3$-dimensional cases I don't see so directly how can one extend this approach. We can restrict to analyze Jordan forms because similar matrices will give isomorphic modules, hence for example in dimension $2$ for $\mathbb{C}$ we have the following cases,
$$ m_T = (X-\lambda) \quad\rightarrow\quad V \simeq \left(\frac{\mathbb{k}}{<X-\lambda>}\right)^2 \simeq (\mathbb{k}_\lambda)^2 \\ m_T = (X-\lambda)^2 \quad\rightarrow\quad V \simeq \frac{\mathbb{k}}{<(X-\lambda)^2>} \\ m_T = (X-\lambda)(X-\mu) \quad\rightarrow\quad V \simeq \frac{\mathbb{k}}{<X-\lambda>} \oplus \frac{\mathbb{k}}{<X-\mu>} \simeq \mathbb{k}_\lambda \oplus \mathbb{k}_\mu $$
whereas for $\mathbb{R}$ and $\mathbb{Q}$ we have an extra case where $m_T$ has degree $2$ but is irreducible, giving $V \simeq \mathbb{k}[X]/(m_T)$.
Can we say something more about the second and last cases, in terms of copies $\mathbb{k}^d$ with certain multiplication structure as it can be done for the other cases? Moreover, is there any way to bypass such a case by case analysis so that the $3$-dimensional setting does not become so cumbersome?