At the Council of Elrond Gandalf recounts his meeting with Radagast near Bree:
'Gandalf!' [Radagast] cried. 'I was seeking you. But I am a stranger in these parts. All I knew was that you might be found in a wild region with the uncouth name of Shire.'
'Your information was correct,' I said. 'But do not put it that way, if you meet any of the inhabitants. You are near the borders of the Shire now.'
'I have been told that wherever they go the Riders ask for news of a land called Shire.'
'The Shire,' I said....
FR 2.ii.256-57, emphasis original)
Why is the name 'Shire' uncouth? The OED shows nothing in the history of the word to suggest the least whiff of disapproval. Even setting aside all thoughts of The Hobbit films, Radagast lives in Rhosgobel near southern Mirkwood, which hardly seems likely to be a hub of urbanity. Be that as it may, it is the name Shire the wizard is commenting on. To be sure, Radagast also calls it a 'wild region,' but again he lives in the Wilderland, himself. Gandalf does not seem to be offended by the description, and indeed he calls it correct, but cautions Radagast about the reaction hobbits might have to hearing the Shire called uncouth. For, as the narrator (Frodo) tells us elsewhere '[t]he Shire-hobbits referred to those of Bree, and to any others that lived beyond the borders, as Outsiders, and took very little interest in them, considering them dull and uncouth' (FR 1.ix.150). Ironically, given that Buttebur calls Frodo and company 'Outsiders -- travellers from the Shire' and instantly apologizes for doing so, the Breelanders clearly have the same opinion of Shire hobbits as Shire hobbits have of everyone else (FR 1.ix.154)
Tolkien is here again having a bit of fun with words, as we've seen him do before. Our word 'uncouth' descends from the Old English 'uncúþ', which literally means 'unknown', and therefore 'strange.' The Shire being unknown to Radagast and the Black Riders, they omit the definite article, because the definite article is used for things that are known. Gandalf of course knows The Shire very well, as do its inhabitants, and so they include the article. To them indeed it is the one and only.
So for Radagast and the Black Riders 'uncouth' has its original sense; for the hobbits it has the more modern sense. Gandalf knows them both.