Is there a notion of a “model category which admits left Bousfield localization?” 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 I do have one example (from the person who asked the question), but it's from unpublished work so I'm not going to write about it here.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?

Most importantly:

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?

 A: One relevant reference, though not answering any of the original questions, is a paper by George Raptis On the cofibrant generation of model categories, where it is shown that under Vopenka's principle, every cofibrantly generated model category is Quillen equivalent to a combinatorial model category. Jiry Rosicky proves the converse direction in Are all cofibrantly generated model categories combinatorial?, so this question is set-theoretical.
An example for your second question may be found in the article by Emmanuel Dror Farjoun Homotopy theories for diagrams of spaces, where a non-cofibrantly generated model category is constructed, which I learned to localize in my theses. It admits functorial localizations with respect to any set of maps and even with respect to some particularly nice classes of maps. I do not have a general theory of left Bousfield localizations for such categories, but it can be often constructed from the functorial localization by Bousfield-Friedlander technique. 
By the way, the localization with respect to a set of maps may not be expressed as a localization with respect to a single map in a general model category. See our paper with Carles Casacuberta The orthogonal subcategory problem in homotopy theory for a simple counterexample.
A: Here is another family of examples of non-cofibrantly generated model categories which admit left Bousfield localization with respect to certain classes of maps. Let $C$ be a proper simplicial model category and consider the strict model structure on $Pro(C)$ defined by Isaksen in 
http://arxiv.org/abs/math/0108189. 
Let $K$ be any set of fibrant objects in $C$. A map $f : X\to Y$ in $Pro(C)$ is called a $K$-local weak equivalence if it induces a weak equivalence 
$$Map_{Pro(C)}^h(Y,A) \to Map_{Pro(C)}^h(X,A)$$
for all $A$ in $K$, where $Map_{Pro(C)}^h$ denotes the derived mapping space in the model category $Pro(C)$. 
Then it is shown in Theorem 4.4 of 
http://arxiv.org/abs/math/0403451
that the underlying (essentially levelwise) cofibrations and the $K$-local weak equivalence define a (left proper simplicial) model structure on $Pro(C)$. This is a left Bousfield localization of the strict model structure on $Pro(C)$, which is not cofibrantly generated (and also not fibrantly generated). In fact, this is the left Bousfield localization of $Pro(C)$ with respect to the class of $K$-local weak equivalences (which is a certain proper class of maps defined using the small set $K$ of objects).
