Let $X$ be an algebraic variety, $S \subset X$ its singular locus, and $x \in S$. Say that $x$ is good if for any neighborhood $U$ of $x$, any top differential form $\omega$ on $U \setminus S$ and some resolution of singularities $U' \rightarrow U$, the pullback of $\omega$ to $U'$ is regular.

An obvious necessary condition is that $S$ is of co-dimension at least 2. An example of a good point is $(0,0,0)$ in the variety defined by $x^2+y^2+z^2=0$. An example of a singular point which is not good is $(0,0,0)$ in the variety defined by $x^4+y^4+z^4=0$.

Here is the question:

Is the notion of good point related to other properties of singular points?

We are mainly interested in the case when $X$ is complete intersection.