The rigidity of quotient singularities in dimension greater or equal than $3$ was established by Schlessinger in his paper *Rigidity of quotient singularities*, Invent. Math. **14** (1971). Roughly speaking, he proved that if $(X, \,x)$ is a local scheme with an isolated singularity at $x$ and $\dim X \geq 3$, then deforming $X$ is equivalent to deform the punctured spectrum $U=X-x$.

More precisely, the key result in the paper is the following

**Proposition.** Let $X$ be a geometric local scheme such that $X$ has an isolated singularity at his closed point $x$. If $\textrm{depth}_xX \geq 3$ then there is an isomorphism
$$T_X^1 \cong H^1(U, \, \Theta_X),$$
where $U:=X - x$. More precisely, under these assumptions the formal deformation theories of $X$ and its punctured spectrum $U$ are equivalent.

Now, assume that $X=Y/G,$ where $Y$ is smooth and $G$ is a finite group of automorphisms acting with an isolated fixed point $y \in Y$. Let $p \colon Y \to X$ the quotient map and $x=p(y)$. Then $\textrm{depth}_xX \geq \textrm{depth}_yY$ and, if $\textrm{depth}_yY \geq 2$, one has $\Theta_Y=(p_*\Theta_X)^G.$

From this, if $\textrm{depth}_yY \geq 3$, it is not too difficult to show that $H^1(U, \, \Theta_X)=0$, hence using the Proposition one obtains $T_X^1=0,$ that is $X$ is rigid.

You can read Schlessinger's paper for further details.