At a conference not too long ago I gave a talk on (left) Bousfield localization and was asked an interesting question afterwards. The question was whether I knew any examples of model categories which were not left proper, but which did admit Bousfield localization. There are some trivial examples, but I'm interested now to hear if anyone knows any interesting or surprising examples in this vein. I do have one example (from the person who asked the question), but it's from unpublished work so I'm not going to write about it here. Since localization with respect to a set of maps is the same as localization with respect to their coproduct, I'll just focus on localization with respect to a single map $f$.
Are there interesting or surprising examples of model categories which are not left proper but which admit left Bousfield localization with respect to every map $f$? What about if we restrict to some non-trivial class of maps $f$?
Also, this question got me thinking of whether one can do Bousfield localization without assuming the model category $M$ is cellular or combinatorial. It has always seemed to me that these hypotheses are there just to make the proofs work (cellularity for Bousfield's original proof and combinatoriality to simplify things even further). Because the proof that the localized model structure exists (i.e. that it is a model category) always comes down to the Transfer Principle for transferring structures across an adjunction, so there is always some drawn out analysis of pushouts. Perhaps because of this there is no hope of removing the hypotheses, but I still want to ask to see if that's what the experts think.
Are there interesting or surprising examples of model categories which are neither cellular nor combinatorial but which admit left Bousfield localization with respect to every map $f$? Or has someone studied this question and found subclasses of maps $f$ which work?
Has anyone tried to classify model categories which admit Bousfield localization? Are there any theorems of the form “if $M$ admits Bousfield localization with respect to a set of maps $S$ then some other thing holds” in the literature?