Malt Shop

Brian (Manager of The Greasy Fork): Britta, you promised that this private party would be done by eight. You're doing this during the peak business hours for nostalgia-themed diners.
Annie: Oh, is that true? Has that been researched?

Brian: It has, Annie. After they get frightened by the evening news, many seek the comforting foods and soothing music of a pre-racial America.

—Community

Obligatory bit of 1950s Popular History. All the kids hang out here and listen to the jukebox play Nothing but Hits. Carhops are optional. The Malt Shop may also be referred to as a diner, a soda fountain, a drug store or a café but the general look is always basically the same. In a Time Travel story ending up in The Fifties, often the first place gone into after the "Mister Sandman" Sequence (as in Back to The Future, for example).

The Greasy Spoon is the Malt Shop's less romanticized (but equally conventionalized) cousin. Watch out for the teenagers who are Sweetheart Sipping or the possible eruption of a Diner Brawl.

Examples of Malt Shop include:

Comic Books

Pop Tate's Choclit Shoppe in Archie.
The Retro Universe of Fawcett City in DC Comics' Power Of Shazam includeed a malt shop (possibly the same one from Captain Marvel's initial Whiz Comics appearances). It even played a plot point in the recent Black Adam miniseries.
Kitty Pryde and Storm bond over milkshakes at The Malt Shoppe in Uncanny X-Men #130.

Film

Mel's Drive-In from American Graffiti
Lou's Café from Back to The Future
Naturally, one of these shows up in Pleasantville.
One is included (free of charge) in the 1957-set Indiana Jones and the Kingdom of the Crystal Skull.
A futuristic version of one of these appears in Attack of the Clones, complete with a robot waitress and a fat Italian-American-sounding alien in an apron in the kitchen.
- George Lucas has stated that this is an homage to his earlier American Graffiti.
Parodied in Top Secret, when Nick Rivers and the Resistance fighters end up in a 1950s-style malt shop...in the middle of East Germany. Includes a dance number to jukebox music.
The film of The Music Man sets one of its scenes in an ice cream parlor. The movie is from 1962, though the musical is set 50 years before that.
Things to Do in Denver When You're Dead is narrated by a character in a malt shop, which also plays a minor role in the story.
This is where Smith and the Hooker with a Heart of Gold meet up at the end of Shoot'Em Up.
The Malt Shoppe is the main hangout of the brainwashed honor students in Disturbing Behavior, with one of the characters pointing out the anachronistic nature of the location.
In Pulp Fiction, Vincent takes Mia to Jack Rabbit Slim's, a nostalgia restaurant designed to resemble a 50's diner/malt shop.
Naturally, a setting for several scenes in Grease.

Live Action TV

Arnold's from Happy Days
The Pie Hole in Pushing Daisies resembles one of these, in keeping with the show's Retro Universe aesthetic.
The Groovy Smoothie in iCarly is a modernised version, used as the Local Hangout. Slightly subverted by the fact that it is frequently mentioned early on but doesn't actually appear until halfway through the second season.
Parodied on a Mad TV sketch with a "nostalgic" diner which for the sake of period accuracy refused to serve black customers.
Bizarrely, the first round of time travel on Supernatural, where Dean sees his youthful parents-to-be on a date to one of these...in 1973. Possibly the influence of her protective father.

Music

"...And I burned down the Malt Shop where we used to go just because it reminds me of you!"

Theatre

Shows up in the musical Reefer Madness, which is technically set in The Thirties but has more of a 1950s feel in general.
Not shown in the opera Street Scene, but discussed in a sextet in praise of the drug store and its ice cream.
One of the later scenes in West Side Story is in Doc's drugstore. The jukebox plays the same "Mambo" heard at the gym dance.

Western Animation

The Scooby Doo gang visited these quite a bit in parts of the plot that would normally have taken them to a bar. Also, the Scooby gang would often start an episode in a malt shop (in the 1969-80 show), when they weren't on a beach.
Futurama had this in their time travel episode to 1945.
An episode of Family Guy features a restaurant with a retro 50s malt shop motif. Cleveland, being black, was not allowed to enter.
Seen in the Schoolhouse Rock segment "A Noun is a Person, Place, or Thing" as the narrator engages in some Sweetheart Sipping with her "best friend."

Real Life

Johnny Rockets, a national chain of sit-down hamburger restaurants with a 1950s motif, including waitpersons with paper hats and checkered aprons, chrome finish on everything, and 5-cent jukeboxes stocked with period pop music.
- They've kept the visuals, but the music's all over the place now, including stuff like the B-52's "Love Shack" and Jimmy Buffett's "Margaritaville".
Sonic Drive-In is a chain of drive-in diners that began in the '50s and has survived and thrived into the present day, complete with carhops who wear roller skates. In Northern cities, they're known primarily for the fact that they advertise even though the nearest Sonic may be tens or even hundreds of miles away (Northern weather isn't that conductive to drive-in diners).
In modern times, malt shops as such are pretty rare—you're more likely to see kids hanging out at the local ice cream shop (admittedly essentially the same thing) or Starbucks.
The Classic 50s is a one-of-a-kind drive-in in Norman, Oklahoma that is more or less identical to Sonic except even more nostalgic of the 1950s. Kids meals even come in cardboard replicas of classic cars!
There is one down in Hillsboro, OR or it was there last time I visited. That area is the sort of place where specialty businesses crowd around.
The A&W's chain of fast-food joints have become known for marketing based on nostalgia: They base many of their commercials, and even the interior on the "classic fast food joint" look of The Fifties. Of course, it's not a Malt Shop per se, but it has many elements of one, including ice cream/root beer floats and nostalgic black and white photos on the walls.
Route 66 lives and breaths 50's nostalgia, so there are plenty of old drive-ins and malt shops along the route to cater to tourists.
Nifty Fifty's in Pennsylvania and New Jersey deliberately recreates the malt/sodashop aesthetic, right down to soda jerks who'll mix a custom soda for you from your choice of a couple dozen syrups and seltzer water.