As someone in the US - I have absolutely zero recognition of the town of Oregon City. All I know about the Oregon trail is a bunch of people died from starvation and dysentery

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

Not a city as much as part of a city, but Coney Island is pretty well known. I was recently speaking to someone in Colombia and even they knew of it!

(I'm part of the sideshow cast there ☺️)

Regina, the city that rhymes with fun!

The village "Wacken" is well known in Germany because they hold one of the worlds largest anual Heavy-Metal festivals. They have a population of around 2000, the festival regularly attracts around 80,000 people.

How about: name a country and name the smallest city you recognize from there. Like New Zealand you could say Rotarua

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.

In Denmark it's probably Snave (No English Wikipedia page for it). Population is a whopping 211 people. It was popularized in a series of commercials for a Danish cellular carrier. The concept was so popular that there even was a movie. I haven't seen it, but the reviews seems to suggest it could be fun to watch... If you are drunk enough.

The word "snave" in Danish can somewhat be translated to snogging in English. Heavy kissing. Which has led to the city having massive problems with theft of their signs.

Hallstatt - Austria

The city so beautiful that the Chinese copied it.

What do you consider small? A lot of people know Cupertino California because Apple are based there, but it's only got a population of 57k. It's arguably more recognizable than the closest major city (San Jose), which has a population of nearly 1 million.

Lajitas, Texas, which once elected as their mayor a goat that drank beer, has a population of 75.

Terlingua, Texas, as made famous by Jerry Jeff Walker and home of the Terlingua International Chili Cook-off, has a population of 78.

Luckenbach, Texas, as made famous by Waylon Jennings and Willie Nelson, has a population of 3.

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.

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

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.

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

Edit: I just realised the question was recognising the name of the city, not recognising city based on a picture...

Probably Svolvær/Lofoten with a population of ~4700. It doesn't have the official status of "City" in Norway 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.

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

World recognition or in-country?

If world I'd guess Huskvarna (pop 24 000) for Sweden.

Known for having a company making chainsaws (among other things) named after it (Husqvarna). It is one of the two main brands of professional chainsaws (the other being Stihl).