  In this paper, we study the cyclicity problem with respect to the forward
shift operator $S_b$ acting on the de Branges--Rovnyak space $\mathscr{H}(b)$
associated to a function $b$ in the closed unit ball of $H^\infty$ and
satisfying $\log(1-|b|)\in L^1(\mathbb T)$. We present a characterisation of
cyclic vectors for $S_b$ when $b$ is a rational function which is not a finite
Blaschke product. This characterisation can be derived from the description,
given in [S. Luo, C. Gu, S. Richter, Higher order local Dirichlet integrals and
de Branges--Rovnyak spaces, \emph{Adv. Math., \textbf{385} (2021), paper No.
107748, 47], of invariant subspaces of $S_b$ in this case, but we provide here
an elementary proof. We also study the situation where $b$ has the form
$b=(1+I)/2$, where $I$ is a non-constant inner function such that the
associated model space $K_I=\mathscr{H}(I)$ has an orthonormal basis of
reproducing kernels.
