This is one of my favourite urban issues to discuss with local friends, and to learn about other cities. I've gone down a deep rabbit hole regarding Cabbagetown in Toronto, for example. I also love how older parts of a city have clearly-defined neighbourhoods, and in newer areas they tend to be relatively meaningless and generic - just the tree-based name of the subdivision, for example.
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For St. John's, I'll just explain some of my favourite ones in or very near the core:
Rabbittown is my neighbourhood. It's working class, homes are often dilapidated, lots of newcomers, lots of public housing, lots of rentals, lots of businesses (including the city's only black barbershop). There are old women with rain hats, druggies, and domestic disturbances. My street, for example, was recently in the news for a shooting and police noted they've been there 100+ times this year. Rabbittown is on the crest of the first hill up from the harbour, with Merrymeeting Road dividing it in half. Downhill toward downtown is rowhousing, downhill away from downtown is a mix with a lot of small, post-WWI cottage-style homes. It's an entertaining place to live with a good sense of community. It even had its own theatre company. The neighbourhood pub is the Peter Easton, named for an infamous local pirate.
Georgestown is directly across Bonaventure Avenue from Rabbittown (east). It was the city's first suburb and indistinguishable from the residential rowhouse areas of downtown. It's very gentrified, artistic, colourful. There's a bakery, a cafe, book store, etc. Lots of young professionals, lots of young families, lots of the LGBT crowd. The neighbourhood pub is the Georgestown Pub.
Shea Heights is an infamous one. It used to be a shanty town and after Confederation we were told to either provide public services like water and electricity, or get rid of it. For some reason, we chose the former. The older homes, almost always with nice expansions, still have a bit of shack at their core. In most ways it's a separate town from St. John's. There's even a steep switchback road to get up there. The Circle is similar - it's where public housing blocks were built for residents of the Central Slum when it was bulldozed in the 50s-60s. Used to be infamous, and growing up there would condemn you to a life of poverty. These days, there are LOTS of immigrants with higher standards than the locals and it's much better-kept and with an improving reputation. My neighbourhood is definitely universally seen as worse these days.
The Battery and Signal Hill are postcard St. John's, a well-preserved 17th-Century fishing village in the heart of the city. Narrow, cliffside roads, colourful homes, all of that. Thick accents among the handful of locals who remain, as well as some of the most expensive homes in the city. Lots of Air B&Bs these days.
Quidi Vidi (including The Gut) is a larger version of the same thing, an old fishing village engulfed by the city but still keeping most of its charm. However, modern subdivisions ring it on almost all sides. This is one of the main flashpoints in St. John's between long-time residents and new residents (including Newfoundlanders)/developers. For example, it's no surprise this is where the video of a woman screaming at an Asian brewery owner to go back where he came from was filmed (and fittingly, his family has probably been here longer than hers - generations).
The rest in the core are mostly just varying degrees of each of the above. Riverhead, for example, has the same class of people as Rabbittown and Georgestown mixed, and the housing stock is of a quality somewhere between the two. That sort of thing.