The Illegible

A character depicted with terrible handwriting. Not just a couple of words that can't be made out here and there, but almost every word will be impossible—or almost impossible—to read.

This does not apply to depictions in animation where everybody's writing looks like random squiggles, or to small children learning how to write. This is when it is explicitly stated by a character that the writing looks like random squiggles or chicken scratch.

This attribute is often associated with doctors, lawyers and other professionals, as well as English teachers and secretaries.

Often Truth in Television. Not to be confused with The Unintelligible, which deals with a character that cannot be understood when speaking.

Examples of The Illegible include:

Anime

In Ranma ½ this is a characteristic of Happosai. His handwriting (in Japanese, of course) is horrible and barely legible. So much that once, even he couldn't read his own handwriting.
Kenshin. Seeing as he was born in a family of farmers, spent about a year as a slave, was raised by a hermetic swordsman from ages six to thirteen, and then left to join a militia, there probably wasn't much opportunity to work on handwriting.
The American dub of Sailor Moon badly Americanized the series as a whole, but actually called the Japanese symbols written by Sailor Moon (Tsukino Usagi) herself "squiggly handwriting" in one instance where the viewers got to see what she'd written (when it wasn't shown, characters would instead comment that she couldn't spell very well or used horrible grammar). In a way, this is actually faithful adaptation, because in the original Japanese version, Usagi has legitimately lousy handwriting in her native language. She either fails to use kanji, or uses the wrong ones.
In Inazuma Eleven, Endou Daisuke's handwriting is outright shown to be completely illegible, and at least one character initially mistakes it for some elaborate secret code. The only other person who could read it is his grandson Endou Mamoru, and even this comes as a big shock to everyone else at first. Later, Kudou Fuyuka comes along and can also read it; this turns out to be foreshadowing another plot point much, much later down the road.

Comic Books

There was an Uncle Scrooge comic that had Scrooge finding an old IOU and being unable to make out the signature. He starts tracking down people he thinks it could be, but finds that he owes each of them money. He eventually realizes that the signature is his own and it was an IOU he wrote to himself.
In Watchmen, the police are unable to tell whether Rorschach's journal is written in some elaborate code or it's just that his handwriting is totally illegible.

Fan Works

Subverted in the Harry Potter fic Where Shadows Go, when the partner in Snape's Healer practice joked that Snape couldn't be a "real doctor" because his handwriting was too legible.
Isaac from Emergence has handwriting so bad that, combined with his severe case of Rouge Angles of Satin, his written communications only barely resemble language of any kind.

Film

In possibly the most memorable scene from Woody Allen's Take The Money And Run, Virgil Starkwell's bank holdup is stalled when the tellers can't make out his robbery note:

Bank Teller #1: Does this look like "gub" or "gun"?
Bank Teller #2: Gun. See? But what does "abt" mean?
Virgil: It's "act". A-C-T. Act natural. Please put fifty thousand dollars into this bag and act natural.
Bank Teller #1: Oh, I see. This is a holdup?

Literature

Hagrid from Harry Potter has a primitive, childish handwriting due to the lack of education.
Dave Barry in Cyberspace claims that, though people learn to write neatly in "cursive" in childhood, their handwriting degenerates into random marks as they grow older. The typical example provided is a scribble allegedly written by President Harry S. Truman on August 3, 1945, intended to be an order of waffles for breakfast but interpreted as, "Let's drop the atomic bomb."
It has been pointed out by Sophie and other characters that Howl cannot write in a way that another human being could read.

Live-Action TV

On Roseanne, the Tom Arnold character Arnie was Put on a Bus and Roseanne showed his note to Dan.

Dan: Who are the Allens and why are they out of spice?
Roseanne: Aliens, Dan, from outer space.
Dan: ...Oh. Well, that makes more sense.

On Seinfeld, Jerry once woke up from a dream laughing about something he scribbled down on a note and went back to sleep. He couldn’t read it in the morning and kept asking practically anyone he met what it said, and each person was certain it said something else. In the end, it turned out it read ‘Sigmund’s flaming balls’. At this revelation, Jerry exclaimed, ‘That’s not funny at all!’
Curb Your Enthusiasm riffs on the old "illegible doctor's handwriting" concept. Larry spends the night with a doctor, which is oddly reminiscent of a doctor's exam from start to finish. In the morning, she leaves him a note that he can't read. He eventually realizes that the only people who can read doctors' notes are pharmacists, so he takes it to a pharmacist to read it to him, which causes a comical misunderstanding.
Chris Kattan's recurring character Suel Forrester on Saturday Night Live is The Unintelligible, so people sometimes ask him to write down what he's saying, but his writing is no more comprehensible than his speech.

Newspaper Comics

This was a Hagar the Horrible strip once. Hagar, who's illiterate, wishes he could read the prescription a doctor gave him. So does the pharmacist (or whatever he was, since he's in the Viking age).
In one Blondie strip, Dagwood's handwriting is said to be so bad that a banker addressed him as "Doctor" when he left.

Radio

In I'm Sorry I Haven't a Clue, Humph would rag on the producer Jon Naismith for his terrible handwriting, at one point claiming he needed to pin it up on a board and run past it to have any chance to understand it.

Theatre

Roman cursive was so notorious for its illegibility that it was joke fodder for ancient playwrights.

Video Games

In the final case of Ace Attorney Investigations, Larry Butz's bad handwriting turns out to be a plot point.
In one of the puzzles in Freddy Pharkas: Frontier Pharmacist, a prescription written by the habitually-drunk town doctor was completely illegible until you used an empty whiskey glass on it, prompting Freddy to remark that it must've been written while looking through the bottom of a whiskey glass as well.
Gaius has a Secret Diary in Rune Factory 3 that you can try to read, but you won't succeed. If he happens to catch you at it he'll admit that he can't always read it either.

Web Comics

In El Goonish Shive Mr. Edward Verres has "Ink Blot handwriting", which isn't quite impossible to read, but no two people ever managed to read in the same way any part of his message, except its header in big letters. On the next page one of viewers muses aloud:

Justin: Maybe it's a map?

ARG! had a prescription.

Western Animation

Homer Simpson: Principal Skinner showed him an "obviously fake" parental note that Bart had turned in, but discovered that Homer's handwriting really was that poor.
An episode of The Weekenders featured Carver attempting to send notes to Chum Bukkit [a band] in various ways. They read these as obscure new lyrics.
Doug once lost his ~~diary~~ journal in which he writes the events of every episode. It's picked up by Roger, seemingly the worst person who could find it, but Roger gives it back unread, simply because he can't make out a word of Doug's handwriting.
Chowder in the episode "Gazpacho Stands Up". Mung tells him to practice his penmanship, and after doing so, congratulates him... for somehow making it worse.
Dan Vs. goes back and forth on this. Various characters have stated that Dan's handwriting is horrible, yet any time his writing actually shows up, it's perfectly legible. Lampshaded (probably) in "The Salvation Armed Forces," where Chris reads a letter from Dan, including the illegible parts ("...even thrumyurg betrayed me, with your cowardice and flurgle..."), only for the letter to show up on screen, with the "illegible" parts clearly written out.

Real Life

Doctors are notorious for having illegible handwriting. There are studies that show doctors do not write any more or less legibly than other professionals, but the consequences for illegible writing is much more severe. This is one of several reasons why there's a strong push for computerized medical records, charts and prescriptions
There's a condition called "dysgraphia" which results in illegible handwriting without necessarily affecting intelligence or ability to read. Most dysgraphics are able to adjust by typing whatever they need to commit to paper, but this isn't always an option, forcing them to attempt to write by hand.
Anyone who starts to fall asleep in class while still trying to take notes has probably ended up with this.
The infamous note Oscar Wilde’s lover’s father passed to Wilde at a party.