A Get needs to contain the name of the city in which it was signed by its witnesses.
What is the Halacha where there are neighborhoods? For example, does one write "London" or "Stamford Hill"? "Borough Park", "Brooklyn", or "New York City"? "הרובע היהודי", "העיר העתיקה", or "ירושלים"?
As a practical matter (speaking from personal experience, unfortunately), "Brooklyn" (ברוקלין) is used, regardless of the neighborhood. There are many teshuvos on this topic, mostly regarding the exact wording and spelling (R' Moshe says, for example, Staten Island, on the river Arthur Kill - I'm not citing the Hebrew spelling here).
I believe "city"' means the lowest level of civil administration, which in most places excludes neighborhoods. It is possible that it depends on (as Wendy says) what they put on the letters. The first נפקא מינה that comes to mind is Queens as opposed to Kew Gardens or Far Rockaway.
R Aryeh Kaplan had a fascinating article in the Jewish Observer, December 1976 (reprinted in The Aryeh Kaplan Reader, p. 74, see here) where he discusses the process used to write the first gittin in Monsey, NY involving R Yaakov Kamenetzky, R Moshe Feinstein and others.
He writes about the particular importance of getting the spelling of all names right. He refers to SA Even HaEzer 128 (see e.g., 128:2, which speaks of cities and not areas or buroughs).
R Kaplan describes how one needs a mesorah (tradition) to decide which city name to use, how to write it and how to refer to the landmarks and rivers close to it (if relevant).
As such I would imagine one writes Brooklyn because it was a city until 1898 (Wikipedia here) but London because Stanford Hill is not a city but an area of London (here). In all cases the local beit din would be the holder of the tradition on the exact spelling.
