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We study the local well-posedness of the nonlinear Schr\"odinger equationassociated to the Grushin operator with random initial data. To the best of ourknowledge, 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 largefamily 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. Theproof relies on bilinear and trilinear estimates.

Probabilistic local well-posedness for the Schr\"odinger equation posed for the Grushin Laplacian