Let $Y$ be an affine scheme over a field of characteristic zero. Suppose we have a group $G$ acting on $Y$ and that the subset of $Y$ of points with non-trivial stabilizer is in codimension greater or equal than $3$. Then, by a theorem due to Schlessinger $X:=Y/G$ is rigid that is $X$ does not have non-trivial first order deformations.

I know that a quadric cone in $\mathbb{A}^{3}$ admits non-trivial first order deformations.

Does anyone know an example of a $3$-fold with finite quotient singularities and singular in codimension $2$ admitting non-trivial first order deformations?

Is it true the naive statement: "the dimension of the space of deformations is bigger for a bad singularity than for a mild one"?

For instance is it true that non-canonical singularities are not rigid?

Thank you very much.