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We study the null controllability of the parabolic equation associated with the Grushin-type operator $A={\partial }_x^2+{\left|x\right|}^{2\gamma }{\partial }_{\gamma }^2,(\gamma >0)$, in the rectangle $\Omega =(-1,1)\times (0,1)$, under an additive control supported in an open subset $\omega$ of $\Omega$. We prove that the equation is null controllable in any positive time for $\gamma <1$ and that there is no time for which it is null controllable for $\gamma >1$. In the transition regime $\gamma =1$ and when $\omega$ is a strip $\omega =(a,b)\times (0,1)(0<a,b\le 1)$), a positive minimal time is required for null controllability. Our approach is based on the fact that, thanks to the particular...