In North America
In the United Kingdom, number plates are issued by the Driver and Vehicle Licensing Agency. They do not approve personalised registrations if they contain words which are offensive in any (widely used) language. It should also be noted that UK plates have to match certain very strict letter/number combinations including the following:
Furthermore, many of the letters are fixed: e.g. in the first four above, the 2nd and 3rd letters, and in the fifth, the first two letters, correspond to the original registration district of the car, and not all combinations have ever been used. In the first four, the numbers can be 1, 2 or 3 digits. Vanity plates that exist include:
There is some additional flexibility available by using numbers that resemble letters (eg S / 5), or by using large black-headed screws to fix the plate to the car to fill in a gap. However, the font style, size and spacing is mandated by law, and the police can (and sometimes do) take people to task for being too creative.
Some plates only acquire significance because of particular owners, e.g. COM 1C was formerly owned by the comedian Jimmy Tarbuck, magician Paul Daniels had MAG1C, while the unremarkable 1967 plate BEL 12E is owned by the Belizean High Commission and CHN 1 is owned by the Chinese Embassy.
In the UK, there are a large number of private dealers who act as agents selling DVLA registrations, as well as their own stock - often purchased at auction or from private sellers.

XXX 999
999 XXX
XXX 999 X
X 999 XXX
XX 99 XXX
MON180X (Moneybox)
BEG41T (Beg For It)
AD06 BOX (A dog box) The United Kingdom
In Australia the various states offer personalised plate schemes, with some states having a yearly fee to maintain the cherished number. In the Australian state of Queensland the personalised plate scheme helps fund the Road Safety Activities Fund.
As of 2006, Austria, Denmark, Hong Kong, Latvia, Poland, Iceland and Sweden also allows such license plates.

In other countries

The game show Bumper Stumpers was wholly based on guessing fake vanity license plates for cash and prizes.
In the American film Falling Down, the main character William "Bill" Foster, played by Michael Douglas, is known through most of the film by his California plate serial: D-FENS (as he was employed by a defense contractor).
In the American television program Knight Rider, KITT's vanity California plate read KNIGHT. (His evil mis-programmed prototypes, seen in two popular episodes, read KARR).
In the American television sitcom Seinfeld, Cosmo Kramer was mistakenly issued a set of New York plates that read ASSMAN. It was later determined that they belonged to a proctologist. New York does not allow the use of the word "ASS" on either their regular or vanity plates.
In the Back to the Future trilogy, the DeLorean time machine's California license plates read OUTATIME.
One of George Lucas' early films was a science fiction vision of the future, THX 1138, and it starred Robert Duvall. In a later Lucas classic, American Graffiti, the 1932 Ford deuce coupe hot rod driven by John Milner has a license plate that reads THX 138. At that time California had not changed to seven-digit plates, but this was not really a vanity plate, just a homage to the previous film.
In the TV show Reno 911! several first season episodes open with two cops chasing a car and reading the plate into the radio. When they realize it's a vanity plate and try to figure out what it reads they end up crashing into the car.
In the opening credits of L.A. Law, a California vanity plate reading LA LAW appears on the back of a Jaguar, though in later episodes, the plate is mounted on a Bentley.
In the 1977 movie Smokey and the Bandit, the Pontiac Trans Am driven by Burt Reynolds (the Bandit) has a Georgia vanity plate that reads BAN ONE signifying his CB radio call sign "Bandit One." This was one of the first movies, if not the first, to prominently feature a vanity plate. During the movie, the Sheriff asks a citizen if he saw the plate on the car and the citizen spells out the license plate.
In the 1990 Columbo episode Columbo: Columbo Goes to College the car used by the Criminology Department to reconstruct the murder of Professor Rusk has a vanity plate reading CRMNOLG.
In the 1984 movie Ghostbusters, the Ghostbusters' 1959 Cadillac Miller Meteor Ambulance has a New York vanity plate reading "ECTO-1".
The film Elvira showed a license plate of the protagonist saying KICKASS.
A character in the film Con Air has a licence plate saying AZZKIKR.
The leading character Sarah Tobias in the 1988 movie The Accused, had the plate SXY SDIE.
The character Missy from Bill and Ted's Bogus Journey has the plate MISSY.
The antagonist from the 2001 horror movie Jeepers Creepers has the plate BEATN U (Be Eating You).
In the TV series Frasier, Martin registers his motor home with the plate RDWRER (Roadwarrior).
In the film Ferris Beuller's Day Off, Ferris borrows his friend's car which has the license plate NRVOUS.
In the film Cellular, the feisty lawyer's car has the plate I SUE U 2.
In National Lampoon's Vacation, a girl has the plate LOVEME. Trivia

Australian vehicle number plates
British car number plates
US and Canadian license plates
Danish car number plates
Hong Kong car number plates
Polish car number plates
Swedish licence plates