There are multiple systems of axioms for modal logic. Perhaps, the most influential is Kripke's based on his theory of rigid designation, which is a revival of Aristotelian essentialism (some properties of objects are "essential", and pick out the same objects in all possible worlds, others are not). You can find a survey of objections to it, with references, in the Stanford article. Another popular system is Lewis's modal realism, which is criticized by Lycan in Modality and Meaning, and... by Lewis himself in On the Plurality of Worlds. More recent criticisms include van Inwagen's Modal Epistemology, which argues that modal knowledge we can have is largely trivial, and Felt's Impossible Worlds, which questions cogency of possible worlds semantics.
As for Quine, his objections were more philosophical than technical, they were not resolved, and after Marcus and Parsons many logicians misunderstood their nature. A detailed historical account is given by Tuboly in Quine and Quantified Modal Logic. Quine's primary objection expressed in Reference and Modality was that necessity of statements seems to depend on how objects in them are designated ("nine" is necessarily greater than seven, but "number of planets" is not), which creates problems for quantification into modal contexts. Kripke's modal semantics introduced in the late 1950s enables such quantification, and many authors mistakenly thought that it also solves the original epistemological problem. Kripke however realized that an answer requires making sense of "de re" or "metaphysical" modality that applies directly to objects rather than to ways of describing them.
Kripke attempted such construction in Naming and Necessity by affirming intuitions about properties essential and inessential in modal contexts:
"Suppose that someone said, pointing to Nixon, ‘That’s the guy who might have lost’. Someone else says ‘Oh no, if you describe him as “Nixon”, then he might have lost; but of course, describing him as the winner, then it is not true that he might have lost’. Now which one is being the philosopher, here, the unintuitive man? It seems to me obviously to be the second... If someone thinks that the notion of a necessary or contingent property... is a philosopher’s notion with no intuitive content, he is wrong. Of course, some philosophers think that something’s having intuitive content is very inconclusive evidence in favor of it. I think it is very heavy evidence in favor of anything, myself."
However, experimental studies since showed that these modal/essentialist intuitions are culturally dependent and context sensitive, i.e. answers depend on how a question is asked, see Beebe-Undercoffer's Cross-cultural Differences in Semantic Intuitions. This brings back Quinean doubts about cogency of modal statements, especially in the context of science, which was his primary concern. Reliance on intuition also raises the issues of ambiguity and illicit metaphysical committments, see Cummins's Reflections on Reflective Equilibrium. This was also anticipated by Quine in Two Dogmas of Empiricism, see summary of objections and rebuttals in the IEP article:
"whether Quine or the conventionalist is right, the primary lesson of this section stands, namely, that metaphysical accounts of possible worlds might be mistaken not just in detail, but in their most basic assumptions".