singularity theory - What strict resolutions of singularities are needed?

For S = {rational singularities} (assuming I've understood the statement right) this is an open and presumably hard question. It's something that Koll'ar wanted (see Chapter 12, I think, of Kollar's Shaferavich Maps and Automorphic forms). Essentially this would let you know that if $X$ has rational singularities, then it also has a compactification $bar{X}$ with rational singularities (which is what Kollar wanted it for I think).

On the other hand, if you are doing more MMP type singularities (for example, S = log terminal singularities), then this kind of thing is often possible due to the MMP. See for example BCHM, Corollary 1.4.3 and also KK, Theorem 3.1. (I've been told more general statements exist also, but I don't know a reference).

EDIT1: For S = Cohen-Macaulay, this is also an open question (Macaulayfication is not strict).

EDIT2: Also see the literature on semi-resolutions. While this isn't quite the same thing, it's very closely related (the idea there is to leave the codim 1 singularities alone if they are nice enough, ie SNC or pinch points). This is a fundamental construction in the study of moduli spaces of higher dimensional varities.