  In a conformally flat three-dimensional spacetime, the linearised higher-spin
Cotton tensor $\mathfrak{C}_{\alpha(n)}(h)$ is the unique conserved conformal
current which is a gauge-invariant descendant of the conformal gauge
prepotential $h_{\alpha(n)}$. The explicit form of
$\mathfrak{C}_{\alpha(n)}(h)$ is well known in Minkowski space. Here we solve
the problem of extending the Minkowskian result to the case of anti-de Sitter
(AdS) space and derive a closed-form expression for
$\mathfrak{C}_{\alpha(n)}(h)$ in terms of the AdS Lorentz covariant
derivatives. It is shown that every conformal higher-spin action
$S_{\text{CS}}^{(n)}[h]\propto \int\text{d}^3x\, e \,
h^{\alpha(n)}\mathfrak{C}_{\alpha(n)}(h) $ factorises into a product of $(n-1)$
first-order operators that are associated with the spin-$n/2$ partially
massless AdS values. Our findings greatly facilitate the on-shell analysis of
massive higher-spin gauge-invariant actions in AdS$_3$. The main results are
extended to the case of $\mathcal{N}=1$ AdS supersymmetry. In particular, we
derive simple expressions for the higher-spin super-Cotton tensors in AdS$_3$.
