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)?