Darren Rousar, Sight-Size.com:

First off, Mineral Spirits are a petroleum based distillate. As such it is relatively new to the artist’s scene and was originally used by artists as a solvent to clean brushes. Commercial paint manufacturers (as in house paint) use MS as well. IMO, MS is a stronger ‘thinner’ than turp.

OMS (odorless mineral spirits) is MS but with various aspects of it removed (like sulphur) to make it more nose friendly. OMS cannot dissolve certain medium additives.

Turpentine is a resin distillate from pine trees and is the more traditional artist’s thinner and additive to mediums.

Turpentine, when used to thin pigment for an imprimatura, presents no problems generally. When using turp thinned pigments over an imprimatura it is possible that one could remove some of the imprimatura layer with the brush. If this aspect is bothersome, there are more or less three alternatives. One is the tone your ground. A toned ground has colored pigment mixed within the ground layer itself. As this is suspended in oil it is ‘locked’ in and will not likely come off. Using your medium as your imprimatura ‘thinner’ would lock it in as well but then this underlayer may be too fat. Alternatively (and perhaps not a really great one at that) one could thin their imprimatura layer with retouch varnish.
I am sure this last one will give many pause. 

AFAIK, the only problem with turp or MS when used to thin a paint mixture (either in an imprimatura or the initial layers of the painting) is a thinning of the oil which binds the pigment. This may be a problem only in that the dried paint may not bond well to the surface. But a layer painted like this is a ‘lean’ layer. I am not aware of any studies showing that this is bad for the painting in the long run. Perhaps others are aware of such studies?

Layering, either intentionally or simply via the act of multi-session painting is always done fat over lean when using oil paint. ‘Lean’ here either means paint straight from the tube or more commonly paint thinned with something other than a fat (oil).