Richard Fallon has a paper on this topic here.
During the 2010 Term, in which the Supreme Court decided 85 cases, it received 56 briefs on behalf of groups of self- identified legal scholars or law professors, with at least one such brief being filed in 30 cases, or more than a third of the total.2 In at least seven cases, dueling teams of scholars filed briefs supporting opposing sides.3 By contrast, during the Court’s 1985 Term, the Justices decided 159 cases4 yet received only three scholars’ briefs.5
In my view, participation in scholars’ briefs gives rise to more complex ethical issues than do most of the other sub-roles that law professors play, largely because the signers of scholars’ briefs represent their submissions as offering distinctively scholarly expertise and perspective. But there is of course no complete divorce between the role of teacher and scholar, on the one hand, and most of the other law-related functions that law professors sometimes take on. To be blunt, law professors recurrently attempt to leverage their credibility as teachers and scholars to influence non-scholarly audiences, sometimes for personal gain and sometimes without satisfying the standards on which their scholarly reputations depend. When law professors seek to trade on their academic reputations in the performance of non-academic roles, two sets of moral and ethical issues arise. One involves the extent, if any, to which an implied warranty of scholarly integrity ought to preclude professors from making assertions to courts or to non-scholarly audiences that they would not make in scholarly books or articles. The other set of worries involves the risk that non- scholarly activities—which would otherwise be permissible or even admirable in themselves—might inhibit future scholarly assertion of inconvenient truths.
Dan Markel has more at Prawfs.
Update: Kenneth Anderson writes:
I have had long suppressed questions about why courts should accept these briefs at all, given that they seem to me – my amicus briefs and everyone else’s – just advocacy leveraged by quite specious claims of “neutral” expertise. Meaning by “specious” – the expertise is real, the neutrality is not.
I think this applies equally to briefs about the Second Amendment.
Update: More from Paul Horwitz.