  This paper studies the function spaces $\mathcal{D}(\mu)$ by Richter and
Aleman, and $\mathcal{D}_{\vec{\mu}}$ by the second author.
  It is known that the forward shift $M_z$ is bounded and expansive on
$\mathcal{D}(\mu)$, and therefore $\mathcal{D}(\mu)$ coincides with a
de~Branges--Rovnyak space $\mathcal{H}[B]$. We show that such a $B$ is rational
if and only if $\mu$ is finitely atomic, and this happens exactly when the
corresponding defect operator has finite rank. We also outline a method for
calculating the reproducing 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 expansive with finite rank defect operator. This
investigation provides many interesting examples of normalized allowable tuples
$\vec{\mu}$.
