Let $X=\{X_1,X_2\}$ be the orthogonal complement of a Cartan subalgebra in the Grušin plane, whose orthonormal basis is formed by the vector fields $X_1$ and $X_2$. When $1<p<\infty$, we prove that weak solutions $u$ to the degenerate subelliptic $p$-Laplacian equation $$\triangle_{X,p}u(z)=\sum\limits_{i=1}^2X_i(|Xu|^{p-2}X_iu)=0$$

have the $C^{0,1}_{loc}$, $C^{1,\alpha}_{loc}$ and $W^{2,2}_{X,loc}$-regularities.