In the mountains of Tennessee, they like a smoky sauce;
But over there in ol' Memphis, a dry-rubbed rib is boss.
The folks down in Georgia can't make up their mind.
You can give it to 'em sweet or spicy, and they'll say "that tastes alright."
You might think that South Carolinians are just a little bit off.
Would you believe their barbecue's yellow—with a mustard sauce?
Pork shoulder is the cut of choice in Mississippi
And they pride themselves on barbecue that's totally vinegary.
And Louisiana, them Cajuns,
They like to spice it up. They like to spice it up.
This is our review of barbecue,
In the southern United States.
And when my life is through,
Bury me in barbecue,
But make sure it's vinegar-based,
'Cause you know that slows decay,
And it's the style from our home state—North Carolina! (Solo!)
Mutton is big in west Kentucky, that's a fancy name for sheep.
But in the rest of the state, sliced pork butt is what they eat.
Arkansas is kinda like Georgia, a barbecue meltin' pot,
And when you ain't looking, them good ol' boys'll make that barbecue hot!
Alabama's got the strangest thing I've seen in my barbecue days,
'Cause their barbecue sauce is white, made out of mayonnaise!
Down in Texas, where the long-horn steer roam free,
They love to smoke their brisket—that's a big ol' slab of beef.
And down in Florida . . .
Is Florida a southern State?
No, it ain't.
This is our review of barbecue,
In the southern United States.
And when my life is through,
Bury me in barbecue . . .
People not from the South
Think barbecue means "cookout,"
And that's something they're wrong about.
(Spoken outro:)
Barbecue is NOT a verb, barbecue is NOT a grill.
Barbecue is MEAT prepared in a very special way,
Which varies depending on where you go.
Best Answer
The "big chain" type certainly exists basically everywhere by now. There is a reason why the Big Mac Index is suitable as an economic indicator: you can calculate it for almost all countries in the world, because you can buy a Big Mac in almost all countries in the world.
A second type of "American restaurant" is much harder to find. It is the kind of small diner which serves grilled cheese sandwiches, thick pancakes and other American style food, without being a chain. I have seen this in places with large expat populations, but most Europeans will probably spend their lives without ever having been in one. I can't talk about other continents.
A third category of "American food" would be American homemade food. Chicken pot pie, eggplant parmesan, Southern biscuits, that kind of stuff. I have not seen it served in any restaurant in Europe. I have never seen an "American home food restaurant", nor an "American fine dining restaurant". They could exist, but as I've visited many large European cities and lived in places with a large number of American expats, they are likely to be quite rare, or maybe clustered somewhere I haven't been.
What I have seen more in later years seem to be fancy burger places - they are sometimes chains, sometimes not, but I would distinguish them from fast food chains because they tend to have things like mushroom-and-arugula burgers on sourdough bread. Also, you nowadays see more burgers in restaurants which are some sort of middle tier without special relation to a specific cuisine - more of a nonfrench equivalent of a bistro. Also, Starbucks style chains (including Starbucks itself) are pretty established - while Europe had an extensive coffee culture before them, the type of drinks served there is pretty distinctive.
American packaged food such as marshmallows or pumpkin pie filling can be also found in specialized grocery stores selling American products, and as a seasonal article in large European supermarket chains like Lidl, who tend to have "American week", "Greek week" etc. in rotation. A few selected American products are also found as staples in most stores, such as Snickers bars or Coca-Cola, or are less available but still within easy reach, such as Jelly Belly candy.