Let $G$ be a complex reductive group acting on a complex affine variety $X$ and let $X // G = \operatorname{Spec}\mathbb{C}[X]^G$ be the GIT quotient.

Is there a relationship between the singular locus of $X$ and that of $X // G$?

Of course, $X//G$ can be highly singular while $X$ is smooth. But, for example, I was wondering if (or under what conditions) singular points of $X$ are mapped to singular points of $X // G$.

**Edit.** Spenser's nice comment below shows that the answer to the latter is no. But perhaps a better and more precise question is: If $X // G$ is non-singular at $y$, is there a non-singular $x \in X$ mapping to $y$? In other words, do all fibres of $X \to X // G$ at non-singular points contain a non-singular point of $X$? I'm willing to assume irreducibility or other nice properties.

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