  We study the local well-posedness of the nonlinear Schr{\''o}dinger equation
associated to the Grushin operator with random initial data. To the best of our
knowledge, no well-posedness result is known in the Sobolev spaces $H^k$ when
$k \leq \frac{3}{2}$. In this article, we prove that there exists a large
family of initial data such that, with respect to a suitable randomization in
$H^k$, $k \in (1,\frac{3}{2}]$, almost-sure local well-posedness holds. The
proof relies on bilinear and trilinear estimates.
