I have written much on this site and elsewhere about the political writings of Noam Chomsky, but have no competence to discuss his academic work in linguistics. Fortunately among my regular correspondents are two leading theoretical linguists, Bob Borsley of Essex University and Paul Postal of NYU, whose advice I benefited from especially when writing for Prospect about the designation of Chomsky as the world's top public intellectual. Bob has sent me the further comments below about Chomsky's influence in linguistics, and has kindly given me permission to post them here - OK
Chomsky has been the most influential figure in theoretical linguistics since the 1960s. In that time, all sorts of people have contested his ideas. However, a good many of those critics have little understanding of the ideas and frankly talk nonsense. I discuss one example of this sort of thing here.
The work of those who know what they are talking about varies a lot in scope and significance. The most important is that which develops a broad general critique of Chomsky's work and advances some sort of alternative. Over the last thirty years a number of alternative approaches have been developed, which share some of Chomsky's assumptions but reject others. Particularly important in my view are Head-driven Phrase Structure Grammar (HPSG) and Lexical Functional Grammar (LFG).
The response of Chomsky and Chomskyans to these approaches has been to ignore them, in effect to pretend they do not exist. They have had some impact but they remain minority currents. In recent years, however, a number of linguists who were once quite close to Chomsky have rejected major elements of his approach. An important one is Frederick Newmeyer, who argues against Chomsky's idea that language variety is the result of a set of innate parameters in his book Possible and Probable Languages (OUP, 2005).
Probably more important is the work of Peter Culicover and Ray Jackendoff, especially their book Simpler Syntax (OUP, 2005), which develops a detailed critique of Chomsky's current ideas, showing how their flaws have their origins in his earlier work and arguing for a simpler approach, which has a lot in common with HPSG and LFG. Also important is Jackendoff's work with the cognitive scientist Steven Pinker on Chomsky's ideas about language evolution.
It is too soon to be sure, but I think we may well be moving towards a theoretical linguistics which is less dominated by Chomsky and less flawed.