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.



Deformations of quotient singularities
Regarding your last question, the answer is **yes**, since there are terminal singularities that are *not* rigid.


For instance, in the recent preprint by Taro Sano *[On deformations of Fano threefolds with terminal singularities](http://arxiv.org/abs/1203.6323)* it is shown that any $\mathbb{Q}$-Fano threefold with "ordinary" terminal singularities (the meaning of "ordinary" is explained in the paper) can be deformed to a $\mathbb{Q}$-Fano threefold with only quotient singularities.


Moreover, the Kuranishi space of a $\mathbb{Q}$-Fano 3-fold $X$ is smooth, hence all first-order deformations of $X$ are unobstructed.


