  In this paper we study the Borel structure of the space of left-orderings
$\mathrm{LO}(G)$ of a group $G$ modulo the natural conjugacy action, and by
using tools from descriptive set theory we find many examples of countable
left-orderable groups such that the quotient space $\mathrm{LO}(G)/G$ is not
standard. This answers a question of Deroin, Navas, and Rivas. We also prove
that the countable Borel equivalence relation induced from the conjugacy action
of $\mathbb{F}_{2}$ on $\mathrm{LO}(\mathbb{F}_{2})$ is universal, and leverage
this result to provide many other examples of countable left-orderable groups
$G$ such that the natural $G$-action on $\mathrm{LO}(G)$ induces a universal
countable Borel equivalence relation.
