Suppose we have a collection, $S$, of singularities types and consider a resolution of singularities (this is: a proper birrational morphism $Y\rightarrow X$ such that Y only contains singularities of the types in the collection $S$ and such that the map is an isomorphism over the points of $X$ with singularity types in $S$.).

For example, if $S$ consists only of the smooth points then $S$-strict resolution is just the standard resolution of singularities. If $S$ consists of smooth points and simple normal crossing points $S$-strict resolution exists. If $S$ consists of only the smooth points and normal crossings points then there is no $S$-strict resolution (in embedding dimension at least 3). As you can't resolve a pinch piont $(x^2+yz^2=0)$ without blowing up normal crossing points.

Have you come around situations in which it is needed strict resolutions (excluding the cases of the first two examples)?

**Extension of question**
(I guess this should go here and no in a new question)
It is exactly the same question but this time asking for resolutions that never involve blowing up a center that intersects the S-locus of the total transform.

eg: -)For S= the smooth points the usual resolution is still fine since it never need to blow-up smooth points. If S is the smooth points and the simple normal crossings singularities this is already not known (I think).

Again the question is: Have you come around situations in which it is needed strict resolutions in this sense (excluding the first example in this second part)?