  We answer a question of Calderoni and Clay by showing that the conjugation
equivalence relation of left orderings of the Baumslag-Solitar groups
$\mathrm{BS}(1,n)$ is hyperfinite for any $n$. Our proof relies on a
classification of $\mathrm{BS}(1,n)$'s left-orderings via its one-dimensional
dynamical realizations. We furthermore use the effectiveness of the dynamical
realizations of $\mathrm{BS}(1,n)$ to study algorithmic properties of the
left-orderings on $\mathrm{BS}(1,n)$.
