Correspondence Course

One step above "Do-It-Yourself For Dummies" but a long, cavernous leap below an actual education is the Correspondence School, an educational institute that forwards all its courses by mail.

Course offerings are many and varied. You'll be amazed by what you can learn by mail: magic powers, hero training, how to talk to girls... everything you could possibly want to know and more, all for the price of postage!

Due to the distant and anonymous nature of these schools, they tend to be less than scrupulous; some may well be outright frauds. But every once in a while (well, more often in fictionland), the heroes discover a correspondence course that actually works—usually to the amazement of the professor teaching it!

A common plotline in comedic mystery stories is the kid who's taking a correspondence course in detection—usually getting in over his head because he hasn't gotten important lessons yet. Another common gag is for the course to be in something you probably shouldn't learn by distance—like scuba diving or hot air ballooning.

A more modern version of this are online courses, which replace the mailed lessons by lessons downloaded from a website. Since these can give legitimate diplomas (even for high school and under) and many are done by existing reputable schools, they don't default to a joke. That a character acquired skills this way can still be surprising.

Examples of Correspondence Course include:

Anime

Ah! My Goddess once featured an exorcist named Shiho who was learning exorcism from a correspondence course.

Comic Books

One of the Super Mario Bros. comics had Toad become a reporter via one of these. Wooster, in something of a Deadpan Snarker moment, points out that the entire curriculum consisted of sending them a check.
One Aniceto comic book story has the titular brujo awaken an ancient Egyptian mummy...which speaks fluent Mexican-dialect Spanish. When asked how, "Simple. I took a correspondence course."
Donald Duck's cousin Fethry swears by various correspondence courses and often shows up at Donald's house in order to either practice his newly acquired skills on a "volunteer" or persuade Donald to start a business with him based on whatever he's been taking a course on. The results are predictable. One comic revealed that apparently the "university" responsible for the courses advertises on the back of a cereal box, so we can probably conclude it is not exactly a highly respected institution.
Superman exposed the crooked correspondence school depicted in the trope image. Naturally, the plan for catching these ordinary criminals involved Clark Kent engaging in art forgery, Time Travel, and Superman unwittingly providing the inspiration for masterpieces by Rembrandt and William Gainsborough. Oh, comics....^{[dead link]}

Film

A major plot point in Bedknobs and Broomsticks: Eglantine learns all her magic from a correspondence course, but the school folds before she can learn the final and most important spell. During the first half of the movie, she hunts down the headmaster, Professor Emelius Browne, who, it turns out, doesn't know magic at all (and more importantly, doesn't know the final spell. The second half of the movie is spent recovering it and then using it against the Nazis).
This was used in the live action adaption of the Discworld novel, The Colour of Magic, to close a plothole in the source material. In the book, the tourist Twoflower initially struggles to communicate using a phrasebook, but after that scene has no problem understanding and being understood by others. In the film, it's Hand Waved by saying he learned the local language through a correspondence course, and was only using the phrasebook for Cloudcuckoolander reasons.
Both The Muppet Movie and The Great Muppet Caper feature this exchange:

"Where did you learn to drive?"
"I took a correspondence course."

Old movie, The Fuller Brush Girl:

Sally Elliot: "It only took you a year to finish that correspondence course."
Humphrey Briggs: "Yeah, but that was a six month course."

Waiting for Guffman:

Clifford Wooley: I had a... hankerin' to be an actor when I was a young feller when I got out of the Coast Guard, but I... I went to taxidermy school instead... well, I took a correspondence course.

Music

In the absurdist song "Yo-Ya" by the Israeli rock group Poogy, a man takes a correspondence course in swimming from a "famous lifeguard". When he finally enters the water, he immediately drowns.

Literature

In Harry Potter, Filch is a Squib (a non-wizard born of Wizard parents) who attempts to learn magic by following a correspondence course. Because of the nature of magic in the HP world, we know it's a fraud, but it's all played for laughs.
The Discworld novel Sourcery features a young man named Nijel who's taking a correspondence course in being a barbarian hero.
In P. G. Wodehouse's short story "A Voice from the Past", Sacheverall Mulliner becomes a domineering Strong Man by taking a correspondence course on ~~scientific agriculture~~ developing an Iron Will.
- Bertie Wooster's analysis of Smug Snake Steggles also deserves mention: "Machiavelli could have taken his correspondence course."
Harry Dresden at one point took a correspondence course in Latin, the language official White Council meetings are conducted in. It was... less than successful.
Taken by Edward Bloor takes place Twenty Minutes Into the Future in Florida, where the main character Charity goes to school at a variant of this. It is school broadcast through webcams.
The titular character in Jean Robinson's The Strange But Wonderful Cosmic Awareness of Duffy Moon attempted to make up for his perceived shortcomings by taking "Dr. Louis Flamel's Home Study Course in Cosmic Awareness". Due much more to his can-do nature and knowledge of gardening than to the course, he actually managed to start his own plant-service business and gain the respect of his aunt and uncle and the other kids in the neighborhood.

Live-Action TV

In the first episode of The Drew Carey Show, Drew is trying to make small talk with the guys in his carpool and sees an ad in one guy's newspaper - "Learn veterinary science through the mail!" "Gee, I'd hate to be the dog in that house."
Mister Rogers' Neighborhood: X the Owl took courses (and possibly graduated, eventually) from Owl Correspondence School, which had its own Fight Song.

Owl Correspondence School
Old O.C.S.
We love you best!

In an early episode of M*A*S*H, Radar was taking a creative writing correspondence course.
In The Secret World of Alex Mack, Dave gets A Day in the Limelight of sorts when he takes a "How to Be a Spy" course on tape to find out Alex's identity. Amazingly enough, he succeeds, but decides to keep it a secret in agreement with the course's final lesson: "If you discover a secret, no matter how big it is, it's best to keep it for yourself" (which if you think about it, goes against the very purpose of working as a spy).
In one season of Last of the Summer Wine, there's a Running Gag in which Seymour attempts to run correspondence courses. Nobody applies for them.
Monk took a correspondence course in swimming.

Video Games

The hero's backstory in Quest for Glory is that he saw an ad for the "Famous Adventurers' Correspondence School" in his town and decided to take it. Persons who pass the course receive a heroic cloak and tools related to their class (lockpick kit for a thief, sword and shield for a fighter, magic spells for a wizard). You lose the cloak in later games, but the correspondence school is constantly brought up. You actually visit its headquarters in the final game.
- The Famous Adventurers' Correspondence School was actually Defictionalized by the series creators, launched as The School for Heroes.
In the Rival Schools games, the nameless class rep learned Dan Hibiki's infamous Saikyo style through correspondence courses.
Insignificant communications in Vega Strike include among other insults this:

Privateer to Pirate: So, flying license by correspondence course, I see.

In Command & Conquer: Generals, if you defeat Dr. Thrax in the Generals Challenge, he'll say "Erm, perhaps I shouldn't have gotten my degree from a mail-order college."

Web Comics

In Dan and Mab's Furry Adventures, Wildy becomes a shaman though one of these. Link.

Western Animation

Dr. Zoidberg from Futurama once claimed to have taken correspondence courses on Murderology and Murderonomy.

Real Life

These obviously exist in real life, although the choice of lessons tend to be a lot less colourful here. Famous distance schools include the Art Instruction School (see below) and the Charles Atlas exercise routine. Nowadays, a lot of them are available online.
Charles Schulz, the creator of Peanuts, took an art course from Art Instruction School. He later became an instructor there.
Ben Cohen and Jerry Greenfield took an ice cream making correspondence course from the Penn State Creamery which enabled them to start their ice cream business.
The Open University subverts most expectations by being one of the UK's largest and most well-respected universities (no Oxbridge, but far from embarrassing either)
- In the United States, MIT has its OpenCourseWare project, the goal of which is to make all of the school's graduate and undergraduate offerings available online. The program was the first of its kind and has since been copied by many other institutions.