we have a town called “Fucking” with only a few hundred people living there. the town sign gets stolen once a month

Wales has St. Davids - population c.1,750

Byron bay. Bundaberg. Coon.

In the UK it’s got to be the City of London. Famous for being an ancient city established by the Romans and awash with history, now one of the world’s biggest financial centers with a modern skyline of famously distinctive skyscrapers. It’s home to some world-famous landmarks like Saint Paul’s Cathedral and Tower Bridge, and has a population of about 10,000.

The City of London is not to be confused with London, London, London or London.

Gruyères, Switzerland. 2000 inhabitants. Famous for the famous Swiss cheese of the same name.

Corleone, Sicily, for obvious reasons. Poulation around 10k.

Roswell, NM comes to mind. Tiny and yet most people will think of UFOs when they hear the name.

Back when I was in Russia I’d say it’d be Suzdal, famous for the density of churches and other traditional architecture; or Tarusa, known for that one song that everyone seems to know a couple of lines from, Gorodok (here is a random rendition I found just now), both with just under 10k pop according to the wiki. And, as a bit of a stretch since it’s not a town and most people would call it Solovki, Solovetsky settlement, famous for being a prison, with about 800 people. Also Oymyakon with under 600 people, the coldest settlement on earth if you’re into that sort of thing.

Now in Georgia, I’d say Borjomi with just over 10k pop famous for its water, and Bakuriani (just over 1800 people) for its water and the ski resort. Again a bit of a stretch, but I guess everyone in Georgia at least also knows the ski resort of Gudauri at just under 100 people, as well as mountain resorts of Gomismta and Bakhmaro, both with no permanent residents due to the rough winters. Geography nerds will also be familiar with Ushguli, (arguably) the highest inhabited settlement in Europe, population 220.

PANAMA!

Van Halen, spring break, population 35,600.

Chicken, Alaska. Population: 12
I’d argue that everyone recognises the word “Chicken”.

Fun fact: The settlers wanted to name their town “Ptarmigan” after the birds that were abundant in the area.
But none of them knew the correct spelling, and they didn’t want to embarass themselves.

I legitimately though Oregon was a State, I didn’t know it was also a city.

From my country I’m going to be generous and say Barcelona. Second biggest city here. I doubt next bigger cities are universally known.

Gergovie. The place where we defeated romans 2000 years ago

Here in Illinois is Woodstock, at ~25,600 (2020 per Wikipedia). It was the town where Groundhog Day was filmed. There’s even some small road signs here and there mentioning it for the rare tourist who comes to see it. Smallest place in the state I can think of, though there’s smaller towns that have been used for movies.

Some upcoming off-brand Hallmark x-mas movie will feature local tourist trap town Long Grove IL, pop ~ 8,300 (2020 per Wikipedia). The director grew up near there so knew about it and thought it’d be perfect for his movie.

In the UK and a city? Probably Liverpool and because of The Beatles.

A Town? Well it certainly used to be Lockerbie where Pan-Am flight 103 crashed after a terrorist bombing just before Christmas 1988. It was on it’s way from London to New York.

Probably not known by the younger generations though.

I don’t know about the smallest, but I’ve always thought that Santa Fe, New Mexico has an outsized influence on everything from food to art to architecture and culture. I visited last year and it was much smaller than I envisioned, partly because there are local regulations on building height to keep from ruining the charm of the city.