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.

Definitely Altschauerberg, home of the country's most prolific performance artist

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.

Gergovie. The place where we defeated romans 2000 years ago. Doesn't even exist anymore

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.

Chicken, Alaska. Population: 12
I'd argue that everyone recognises "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.

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.

Ramstein, population ~5600

Famous for the Ramstein Air base, the bombing of the air base, the Ramstein air show disaster and the band named after all of that.

For the US, I'd say a pretty strong contender is Woodstock, NY, with a population of around 6,000, and of course famous for the music festival of the same name (even though the actual festival was something like 60 miles away in Bethel)

PANAMA!

Van Halen, spring break, population 35,600.

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.

In the Netherlands is probably Giethoorn, the 'Venice of the North' which has many canals instead of roads and is very touristy. It has 2.900 inhabitants

Wąchock in Poland, (in)famous for being the place where tons of jokes happen, population around 2800.

Also Jeruzal, though known under its fictional name of Wilkowyje, the place where famous TV show "Ranczo" was made, population around 340.

Not my country, but what immediately came to mind was one that has global name recognition, and minimal population: Chernobyl.

It used to have around 12,000 population, but now it's technically illegal to live nearby, and up to 150 people are estimated to live there today. It's famous for being toxically irradiated as a result of the worst nuclear disaster in human history

Wacken, Germany.

Population: 2110

Home to one of the biggest metal festivals in the world with something between 70k and 120k people. I think Tickets are limited to 70k currently but the whole area is bascially transformed for a week

Schengen - the village in Luxembourg where the Schengen Agreement was signed. The population was 5196 in 2023 (appears to be the last census quoted on Wikipedia) and the "Schengen Area", covered by the agreement represents 450m people.

Source: https://en.m.wikipedia.org/wiki/Schengen_Area

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.