Taeko from Ai Yori Aoshi is praised as a good driver, by the proprietor of an arcade. Tina, Kaoru, and Chika are just relieved to be alive.
Haruko from Air terrorizes the neighborhood on her motorbike.
Every named character in the manga version of AKIRA. Then again, aside from intentionally reckless motorcycle hoodlum Kaneda, we mostly see them trying to control military vehicles they weren't trained for. Special mention goes to Kaneda, Kei and Chiyoko for pulling off an honest-to-goodness tank rampage.
From Air Gear, the character Tomita, the homeroom teacher for the protagonists, drives so poorly that they end up knotted together when they exit. How that's possible, nobody knows.
Subverted in Asteroid in Love. When Mikage mentiones in the twenty-seventh chapter that she is getting a driver's license, Mai and Mira fall silent, as they are having an Imagine Spot about Mikage going this when she drives. Eventually subverted, however, since Mikage drives rather responsibly when she drives the club to Tsukuba in the eleventh episode.
Osaka:[watching from Nyamo's car] They're flying...
After her first ride with Yukari, Chiyo-chan takes measures to avoid becoming Yukari's passenger again, including placing herself in Nyamo's car the very instant she found out that Yukari was the other driver during the third-year trip. In the manga, it's implied that the ride scared the poor girl off riding in vehicles altogether. Hearing this, Kaorin goes into a panic and is reluctant to take the ride in Yukari's car but decides to do it anyway because Sakaki was willing to do it. Yukari's little slip-up in the front yard terrifies Kaorin to the point of Color Failure. She ends up totally shell-shocked by the time they get there.
On the plus side, victims of the Yukarimobile tend to report conquering their fear of roller-coasters.
Kaorin: The nice thing about roller coasters, they don't get into collisions...
Tomo is the only one who actually seems to like Yukari's driving style, but then again, she's Tomo.
Only Isaac Dian could drive a slow moving Model-T in a straight line and accidentally run over four people. Okay, so that first smack against Dallas was intentional, but backing up and parking on him and his buddies was probably not. Good thing for that healing factor.
In the 2001 novel, Maiza of all people is accused of this by his passengers. Said passengers are immortal, and are therefore not likely to be all that upset about run-of-the-mill crazy driving.
Arianna of Campione! drives at highway speed on narrow streets, twice that on highways, and seems completely unaware of how close she comes to killing everyone while driving. Despite this, she has never had an accident.
Hyoma Aoi from Combattler V. He is not a lousy driver but a Badass Biker, but he likes a bit too much having fun as he drives. In his first appearance he caused a traffic accident because he broke the limit speed and several patrol cars were trying to catch him. All of his chasers crashed into each other, but he avoided to crash.
Rosette from Chrono Crusade. One of the first things she does after we're introduced to her is completely total her car. In the anime she destroys several. Even the mangaka doesn't know if she actually has a license.
Jaco isn't usually reckless when flying his ship, but has a bad habit of getting distracted that can kick in at the worst possible moments. Hence why he somehow managed to miss the fact he was flying into the moon until the last moment and got stranded on Earth in his solo series.
No matter what vehicle she's in, Bulma will drive it aggressively, often risking to crash into someone-and in fact, that's how she first met Goku, by running him over with her car. Interestingly enough, she grew somewhat less aggressive with age, as shown when she was on Jaco's ship and he made her feel ill by pulling the same manouvers she had pulled the first and only time he let her drive it when she was five... But she's still aggressive enough she crashes her car into Goku again a few weeks later (this being Goku, the crash ends up hurting her more than it hurts him this time).
Goku himself is no better. Just watch the episode where he and Piccolo try to learn how to drive.
And on that subject, there's also Piccolo's driving instructor. She not only manages to scare Piccolo, who wonders aloud if she even knows what she's doing, but encourages him to race against Goku over bad terrain.
And finally, we have Chi-Chi, if episode 8 and this ending to GT will prove. And she's the one who told Goku to get a license.
Not only did Agon of Eyeshield 21 hijack a car, but by using his God-speed impulse reflexes, he knows exactly when and where to make sharp turns to avoid the sane drivers. Did we mention he hijacked an American car?
Haruko from FLCL has caused more than her share of accidents — on a motor scooter, no less. In one episode, she pulls out in front of traffic which causes the entire lane of cars behind her to spontaneously begin flipping end over end◊ (presumably from slamming on the brakes so hard). Her Vespa is also her means of FTL Travel, so obviously she can go at supersonic speed on it through the town. No problem!
An unrelated Yukari in the manga Family Compo manages to scare the hell out of her entire family while out practicing for the first time in fifteen years. She'd managed to turn into the wrong lane and drove against traffic for a while. Also has a poor grasp of what speed is appropriate.
Fate/Apocrypha: Mordred has the same approach to driving as her "father," Arturia (see below). She has the same B rank Riding skill that makes Arturia a Badass Biker in Fate/Zero, but she drives so recklessly that Kairi commented that nothing short of a tank could survive her driving, and indeed the car in question is covered in dents and looks ready to fall apart by the time she finally gets out (by kicking out the driver door, no less). She nearly slams into Astolfo on the battlefield with it and he only barely gets out of the way in time.
Fate/Zero: Irisviel is not as good a driver as she thinks she is. Since Saber can't really be harmed by physical means and is fast enough to save her if there's an accident, both of them get to enjoy going around 65-90 miles an hour through winding terrain at midnight without ever stopping for petty concerns like traffic lights or being on the correct side of the road. Even Saber has a better grasp of traffic laws and car operations than her, and she's a Knight in Shining Armor from the medieval era. Saber also qualifies in that she has the ability to pilot absolutely anything, including a jetliner, and likes taking potholes and coastal roads at upwards of 60 miles per hour. Unlike Iris, she's actually an inhumanly good driver, which is why she has absolutely no problem with breaking every traffic law in existence (it's not like a car accident can hurt her).
Fullmetal Alchemist gives us Roy Mustang, who apparently is in the habit of looking away from the road while talking to his passengers. This is probably one reason why his staff usually drives for him.
A female cop who decides to chase Sousuke and Kaname (who are riding on a bicycle) down in her police car. Her car flies off edges of the street, rams into barriers and signs, and she is laughing maniacally as her partner is panicking. This scene is just a one long Shout-Out to the Natsumi Tsujimoto being Late for Work in the first You're Under Arrest! OVA. Doubly ironic because there it was only Natsumi who drove that way, but here it is for BOTH parties.
In every instance of Sousuke driving, he's been shown to blatantly ignore rules or almost get into accidents. Most of his driving skills seem to be designed for dangerous car chase missions. Even riding a bicycle is crazy when it involves him.
Ghost in the Shell: Stand Alone Complex: Maj. Motoko Kusanagi. Although most of the time she remains at the level of Badass Driver with a Cool Car, the time when she plowed her vehicle straight through the front doors of a hotel and up a flight of stairs before screeching to a halt where the lounge's coffee table used to be, among other incidents, pushes her into this territory. And that's without getting into the fact that she dives the 'net while driving.
Woolf Einnacle was a professional racer before joining the Federation forces, and is quite open in demonstrating his skills to young Flit and Emily—whether they want to see or not.
Maiya is a Cat Girl from a civilisation that arose from the ruins of an ancient war, and all of their technology is dug out from the dirt with them having to learn how to use it on their own. That said, the hover vehicles appear to be quite common considering how much her people rely on them, meaning that her driving skills are more of this trope than lack of formal lessons.
Rally in Gunsmith Cats. She's not incompetent, indeed far from it, she's just... enthusiastic. As in does-180-degree-skid-into-parking-space-on-other-side-of-the-road-level enthusiastic. This, combined with her tendency to get into high-speed car chases with dangerous criminals, is the reason she can't get collision insurance anymore.
Rally: <innocently as she leaves perfectly parked car> "What's wrong?" May: <staggering out of the passenger seat> "C-carsick...."
A variation in the manga Hana the Fox Girl. Aoi the dragon is, like all of his kind, a natural born flyer. However, he has trouble finding the standard dragon work (such as a carriage or messenger for the gods) because he's absolutely terrible at flying. Hana describes it as like a roller coaster they saw on TV "but with waaaaay more spins and turns!"
Bunta, Takumi's father from Initial D. Especially when he calmly pulls out and lights a cigarette in the middle of a drift. When that happens, Takumi's boss looks horrified.
Mario Zucchero from JoJo's Bizarre Adventure: Vento Aureo. He drives just fine, but like Mustang above, he tends to look away from the road while driving (the most mundane yet most idiotic mistake any driver can make). Because of this, Sale had to act as his navigator during their conversation about Polpo's death and the fortune he left behind. Not only that, but he also leaves the car while it's in motion!
Touko Aozaki from Kara no Kyoukai proves to be a mild example of this in the third movie, judging by Mikiya's reaction to her careening around corners, and in the fifth it's revealed that she never went to driving school. It's also implied that this may be a trait of mages in the Nasuverse in general since most of them are rather odd and many of them abhor modern technology.
While he seems to be a good driver in the games, Sora is shown to be a hilariously bad pilot for the Gummi Ship in the manga adaptation for the Kingdom Hearts series.
In the manga, when Keitaro returns from his time overseas, he's learned to drive — from Seta. Predictably enough he takes after his master, right down to parking his van on its roof. The very first place he crashes, unsurprisingly, is smack dab in the middle of the girls' hot spring interrupting the girls' climactic confrontation with Kanako. Their driving styles are so identical that they just assume it's Seta till he steps out of the car.
Yui in Lucky Star... despite working as a traffic cop! (She was obeying the speed limit, despite the stunts she pulled!) Considering the entire first time we see her drive is an Initial DShout-Out, that is saying something. It's also worth noting that despite driving like a maniac, she proves herself to be extremely skilled in driving, pulling off some truly amazing stunts.
Nurse Youko from Mai Hime. Hearing one single, high-pitched, utterly terrified scream from Midori as Youko drives away should clue you in.
Sei in Maria-sama ga Miteru would be able to give the aforementioned Yukari a run for her money on how she drives. This is even referenced in Maribato!, where Sei appears as a playable character: One of her supers has her driving a car, and attempts to run over her opponent.
Koji Kabuto from Mazinger Z. He is not a bad driver. Oh, no. In fact, when he drives he is the only person on the road who is absolutely safe. He is a Badass Biker who believes traffic regulations are mere suggestions, speed limits are a myth and bikes were made to pull crazy stunts with them.
Hinako from Murciélago, to an insane degree. Aside from your standard insane driving, you can also add to the list driving down train tracks to catch a train and driving on rooftops so she can find Kuroko.
My Monster Secret: While taking the cast on a camping trip, Akane offers to take over driving duties so that Akari can relax and knock back a few beers at a rest stop. Unfortunately, Akane is an ancient demon, and while she technically knows how to drive, her knowledge of automobiles is roughly a century out of date. She resorts to using her telekinesis to move the van. Said van doesn't survive the trip.
Misato is often thought of this way in fanon, but this is something of a case of Never Live It Down because the first time she did it was deliberate — she was out to scare Shinji. She also did it at the start of Rebuild 2.0 when the water-skimmer Angel attacked (drifting at high-speed on the overpass).
To a lesser extent, Mari also fits this trope while using Unit 05 (it has wheels instead of legs).
Mihoshi from the No Need for Tenchi! manga tried to learn how to drive. It did not go well. Then she got in a Chase Scene, and all was forgiven, somehow.
Noike from the Ryo-Oh-Ki OVAs is this squared. The promotional image of her driving a family truck◊ is full of the household members and farming implements spilling out of it, while Noike stays by the wheel with not a worry in the world.
Pani Poni Dash! has Old Geezer, who Becky is lead to believe is the owner of the shiny green MGB parked out on the school grounds. Old Geezer tricks her into taking her for a ride with the car, only for Miss Igarashi to come out and ask who just drove off with her car. The following scenes afterward show Old Geezer driving so fast over Antarctica and nearly hitting a frozen woolly mammoths. Becky's scream could be heard from space, and she understandably develops a fear of cars.
In episode 119 of Ranma ½, the main cast (and a one-off character) take part in a kart race for a full-course dinner. The results are exactly as expected as the majority keep crashing and bashing into one-another, causing a complete pile up. The only two not like this? Nabiki and Kasumi, who took their time.
Aria from Rising × Rydeen whose first meeting with the main character was right after she nearly hit him with her car and her driving has the main character and her tough looking bodyguards vomiting after they reach their destination.
Yuka's mother from Puri Puri Chi Chan is a very good driver....until she gets to a mountain road.
Exemplified with Straight Cougar of S Cryed. Not only does he drive like a maniac, his "special ability" is to essentially transform the car into the most over-the-top dragster in the world. His crazy driving is more the result of him always pushing his driving abilities to as far as they go than him being a bad driver. He's actually a very talented driver. Nonetheless this often leaves Mimori sick to her stomach, if not puking.
Eudial from Sailor Moon S. Her preferred way to show up in front of the Victim Of The Week is pulling an epic Car Fu. She was so bad that it was a very easy thing for Mimete to tamper with her car and get her killed at the end of her mini-arc.
Kanbara from Saki, as revealed in the manga. Leads to a fun scene with Momo clinging to Yumi in fear as Kanbara takes turns at full speed while driving the team's van.
Tomoe from Sasameki Koto. We don't actually see her driving, but afterwards the car looks like it's been in a warzone. And then it explodes.
Sgt. Frog: Aki Hinata has a car as well. She is already known as a terror on the mountain roads in the series and is implied to have traversed miles of implausible terrain in a Citroen 2CV in the fourth movie.
Subverted by Hiyori in Sketchbook, who intends to demonstrate some wild driving to her students, but forgets to disengage the handbrake, resulting in travel down main roads at approximately ten miles an hour at best. She doesn't even realise why they were going so slowly until the group are on the way back.
Space Brothers has Deneil Young who drives like crazy, and also happens to be a pilot instructor who flies like crazy. Most ASCAN candidates try to avoid Deneil Young because nobody can stomach his flying, and everybody hurls on their first flight with him. Oh, and he also practices using a wheel chair (even though he can walk just fine), and he drives like crazy with that too.
For that matter, there is also Vincent Bold, who always drives extremely quickly because he has an obsession with avoiding wasted time. He also takes pleasure in intimidating ASCAN candidates with his driving (and gauging their reaction).
In the Trinity Blood anime, Sister Esther attempts to drive, and nearly gives Abel a heart attack.
Aforementioned chase in the first You're Under Arrest! OVA. Natsumi is really a genius when it comes to anything on the two wheels, but she was really late that morning. And she just couldn't drive a car, where she plays this trope dead straight, to the point that her inability to get a four-wheeled license (and she's a traffic cop, dammit) is a major source of comedy in a couple of episodes.
Her partner and best friend Miyuki Kobayakawa, on the other hand, is with the cars what Natsumi is with the bikes, but have a tendency to the Let's Get Dangerous! situations, and loves to push the pedal to the metal — too bad that she has a little chance to do it.
In Yu-Gi-Oh! GX, Jim Crocodile Cook does this — in a fossilized truck. Season 4 gives us Saiou saving Juudai by running over the Trueman clones with a van' and then escaping on a motorcycle.
In Yu-Gi-Oh! 5D's, being able to do this without killing yourself is a prerequisite for a duelist who uses a D-Wheel, which means most duelists in the series; the faster the D-Wheel duelist drives, the more powerful the Speed Spells he can use, and the top speed can be around 140 MPH. In the second season, when D-Wheels become far more popular, Neo Domino (and presumably other major cities) take serious steps towards public safety, restructuring the highways to adapt for D-Wheel duels, and switching to the new "Speed World 2" card with that in mind.
Its mangacounterparttakes it up a notch at one point: Late in the turbo duel between Aki and Sherry, Aki closes her eyes while still riding on her D-Wheel. Aside from Yusei, who thinks she's truly enjoying her duel, everyone in-universe reacts with shock and horror (one other duelist remarks with: "Forget the ICU! One wrong move and she'll be in the morgue!"). To be fair, however: 1.) She does this as a necessary measure to block Sherry's Hand Scan ability, 2.) she KNOWS how dangerous such a stunt is, which is why she hesitates about doing so, even at the brink of defeat, 3.) it takes some encouragement from her friends and fellow classmates (instigated by one fellow classmate Aki herself defeated in their respective duel, no less) to get her to really let go of her inhibition, and 4.) when asked by Sherry about where Aki learned to ride her D-Wheel so safely despite closing her eyes after Aki wins the duel between them, Aki is so shocked by the realization of what she had done that she loses the strength in her legs momentarily, before admitting to Sherry that it's a desperate measure that she can't believe she managed to pull off and that she can't be counted on to pull off again.