I don't know how much of an answer this is, since I also don't know the extent to which the question can be answered precisely. It seems to me (based mostly on conversations) that most mathematicians (and logicians) waver between a kind of platonism and a kind of formalism, though when pressed the may actually come down in favor of formalism. But this is purely anecdotal, I don't know of any survey which attempts to track down what most of the community actually thinks (this would be an interesting survey, though).
Note that there are many interrelations here between logic and mathematics, so that adopting one position viz. logic will generally force you into a similar position viz. mathematics and vice-versa. For instance, adopting pluralism regarding logic will probably force you into adopting pluralism regarding mathematics, as it is well known that a difference in underlying logic can generate rather different mathematical theories (for instance, smooth infinitesimal analysis is fine under intuitionist logic, but inconsistent under classical logic). On the other hand, rejection of intuitionism may entail rejection of intuitionists mathematical theories. So one needs to strike a careful balance here between your logical and your mathematical sensitivities; since you asked for references, Stewart Shapiro's new book, Varieties of Logic is a spirited defense of pluralism in both domains.
One interesting consideration when evaluating pluralism is the choice of the background meta-logic. As Timothy Williamson remarks in "Logic, Metalogic, and Neutrality", while it's a fact that there is a plurality of logics to work with, it's not so clear that there is a plurality of meta-logics to work with. In particular, most work done on the meta-theory of, say, first-order intuitionist logic takes place in a classical setting. For instance, most proofs of the completeness of first-order intuitionist logic are done by adopting classical first-order logic as one's meta-theory. Similarly, the semantic clauses for fuzzy logic are generally formulated against the backdrop of classical logic. Notice that in all those instances, that does not mean that the choice of meta-logic is merely one of convenience: it's not even clear how one would prove completeness for first-order intuitionist logic using only intuitionism, nor how to formulate fuzzy semantic clauses for fuzzy logics. So the choice of a meta-logic, as Williamson argues, is hardly a neutral issue, and here classical logic definitely has an edge, which may be considered as (strong?) argument in favor of classical logic.
Another issue that can be delicate here involves the primitive notions in a given logic. You disparage the classical logician by pointing out to the many different logical systems, asking how it is possible to hold that there is one true logic in the face of such diversity. Well, one possible reply from the classical logician is that those logics are not really logical at bottom. Take, for instance, epistemic logic, which is generally framed in terms of a modal logic. Why should we consider the epistemic operator as logical? Surely the notion involved (that of knowing something) is, well, epistemic? Similar consideration apply, with even more force, to quantum logic or (say) Montague grammar, whose primitive notions seem to be (close to) empirical. Nothing of the sort happens with classical logic, whose logical primitives generally pass all the tests for logicality. That is not to say that these other "logics" are devoid of interest or somesuch, just that they are not really "logics" in the strong sense of the term. They may serve to model some interesting empirical phenomena, but they don't describe the logical structure of reality, so to speak. Of course, that depends on which criteria you adopt for a notion to be logical in the strong sense of the term, a vexed question if there is one (John MacFarlane surveys some of the options in his "Logical Constants" article on SEP; Gila Sher also has important work in this area -- her book The Bounds of Logic is freely downloadable from her website).
The above debate may also be motivated by metaphysical considerations. An adoption of an extreme form of physicalism may motivate extensionalism, thus ruling out modal logics as being anything but useful fictions (if you're Quine, not even that). It may also favor classical logic instead of intuitionism, since most intuitionist considerations are raised in the context of mathematical problems, which will be meaningless under this theory. Finally, it might also declare things like fuzzy logics uninteresting, since vague predicates will also have no correspondence to anything in reality. On the other hand, the more abundant your ontology, the more you may contemplate something like pluralism. So if you accept the existence of, say, mathematical structures, you may want to say that different logics describe different structures, and that therefore they are all acceptable within their restricted range (so fuzzy logic would be fine for fuzzy structures, classical logic fine for classical structures, etc.). This seems to be similar to Shapiro's position in the book mentioned above; he also defends this in an article in a recent collection edited by Penelope Rush, The Metaphysics of Logic, which has many articles which you may want to check out.
Anyway, these are only samples of current debates surrounding the issues you mentioned. It's definitely not a settled issue, as you can see!