H.W. Fowler, A Dictionary of Modern English (1926):
Their should be his; & the origin of the mistake is clearly reluctance to recognize that the right shortening of the cumbersome he or she, his or her, &c., is he or him or his though the reference may be to both sexes. Whether that reluctance is less felt by the male is doubtful; at any rate the OED quotes examples from Fielding (Everyone in the house were in their beds), Sydney Smith, Thackeray (A person can’t help their birth), Bagehot (Nobody in their senses), & Bernard Shaw. It also says nothing more severe than it is ‘Not favored by grammarians’; that the grammarians are likely, nevertheless, to have their way on the point is suggested by the old-fashioned sound of the Fielding & Thackeray sentences quoted; few good modern writers would flout the grammarians so conspicuously.
Fowler’s Modern English Usage, edited by Sir Ernest Gowers (1965):
Undoubtedly grammar rebels against their; and the reason for using it is clearly reluctance to recognize that, though the reference may be to both sexes, the right shortening of the cumbersome he or she, his or her, etc., is he or him or his, as his and him are used with boldness surprising in a government department in There must be an opportunity for the individual boy or girl to go as far as his keenness and ability will take him. Whether that reluctance is less felt by the male is doubtful; at any rate the OED quotes examples from Fielding (Everyone in the house were in their beds), Sydney Smith, Thackeray (A person can’t help their birth), Bagehot (Nobody in their senses), and Bernard Shaw. It also says nothing more severe than it is ‘Not favored by grammarians’. In colloquial usage the inconvenience of having no common–sex personal pronoun in the singular has proved stronger than respect for grammarians, and the one that is available in the plural is made to serve for the singular too. But in prose their disfavour is not treated so lightly; few good modern writers would flout them so conspicuously as Fielding and Thackeray did in the sentences quoted. …
The New Fowler’s Modern English Usage, edited by R.W. Burchfield (1996):
Over the centuries, writers of standing have used they, their, and them with anaphoric reference to a singular pronoun or noun, and the practice has continued in the 20c. to the point that, traditional grammarians aside, such constructions are hardly noticed any more or are not widely felt to lie in a prohibited zone. Fowler (1926) disliked the practice … and gave a number of unattributed ‘faulty’ examples. …
The evidence presented in the OED points in another direction altogether. From the 16c. onwards they has often been ‘used in reference to a singular noun [or pronoun] made universal by every, any, no, etc., or applicable to one of either sex (= “he or she”)’. … All such ‘non-grammatical’ constructions arise either because the notion of plurality resides in many of the indefinite pronouns or because the absence in English of a common-gender third person singular pronoun (as distinct from his used to mean ‘his or her’ or the clumsy use of his or her itself).
Fowler’s Dictionary of Modern English Usage, edited by Jeremy Butterfield (2015):
Fowler (1926) was among those who objected to the use of their in contexts that call ‘logically’ for his (though this use of the masculine gender to cover both is contentious) or his or her. …
The issue is unresolved, but it begins to look as if the use of the plural third person determiner to refer back to a singular pronoun is now passing unnoticed by standard speakers (except those trained in traditional grammar) and is being left unaltered by some copy editors, though among many it is a bone of contention. Its value lies in its being gender-neutral and avoiding the inherent sexism of his and the cumbersomeness of his or her