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This paper studies the function spaces $\mathcal{D}(\mu)$ by Richter andAleman, and $\mathcal{D}_{\vec{\mu}}$ by the second author. It is known thatthe forward shift $M_z$ is bounded and expansive on $\mathcal{D}(\mu)$, andtherefore $\mathcal{D}(\mu)$ coincides with a de~Branges--Rovnyak space$\mathcal{H}[B]$. We show that such a $B$ is rational if and only if $\mu$ isfinitely atomic, and this happens exactly when the corresponding defectoperator has finite rank. We also outline a method for calculating thereproducing kernel of $\mathcal{D}(\mu)$ for finitely atomic $\mu$. Similarly,we characterize the allowable tuples $\vec{\mu} = (\frac{|dz|}{2\pi}, \mu_1,\ldots, \mu_{n-1})$ such that $M_z$ on $\mathcal{D}_{\vec{\mu}}$ is expansivewith finite rank defect operator. Our investigation provides many interestingexamples of normalized allowable tuples $\vec{\mu}$.

On Dirichlet-type and $n$-isometric shifts in finite rank de~Branges--Rovnyak spaces