The Doctor's scarf

Although the Doctor owned many scarves throughout his lives, the Doctor's scarf from his fourth incarnation was the one he used most. At least two such scarves existed. (TV: Robot, TV: The Leisure Hive) The second was later unravelled by the Fifth Doctor to leave a trail for himself to find his way back to the console room. (TV: Castrovalva)

Profile
Shortly after his third regeneration, the Doctor had to accompany Brigadier Lethbridge-Stewart on an investigation. He tried on several inappropriately conspicuous costumes before settling on a vaguely Bohemian outfit. This included a wide-brimmed fedora and a long scarf. (TV: Robot) The scarf had originally been knitted for him by Madame Nostradamus. (TV: The Ark in Space)

Near the end of this incarnation, the Doctor switched to a scarf of longer length, but coloured in shades of burgundy. (TV: The Leisure Hive) However, his original scarf was kept on a hatstand in the console room. (TV: Warriors' Gate)

After his fourth regeneration, the Fifth Doctor unravelled his burgundy scarf in the TARDIS' corridors, allowing his companions to follow him. After this, he began using pieces of his former self's clothing to continue the trail before finding a cricketer's outfit. (TV: Castrovalva)

The Fifth Doctor admitted that the scarf was warm and sometimes useful, though he did trip on it occasionally. (AUDIO: No Place Like Home)

The Seventh Doctor used the tartan scarf from his newly-found outfit to stall the Rani as he tried to escape her laboratory. (TV: Time and the Rani) The aforementioned tartan scarf was subsequently replaced by a paisley one. (TV: Paradise Towers)

Searching for her rucksack, Ace entered into the console room briefly wearing a long, multi-coloured scarf. (TV: The Greatest Show in the Galaxy

While selecting an outfit following his seventh regeneration, the Doctor discovered a similar multi-coloured scarf in a hospital locker. He did not include it in his outfit, though he did think about using it for a moment. (TV: Doctor Who)

On her first trip back to Earth after being transferred to Destrii's body, Izzy Sinclair wrapped herself up in the original scarf to hide her non-human body. (COMIC: The Way of All Flesh)

The TARDIS wardrobe contained a multi-coloured scarf during the Doctor's tenth incarnation. (TV: The Christmas Invasion)

The Eleventh Doctor and Kazran Sardick wore scarves similar to the Fourth Doctor's when visiting Abigail Pettigrew. (TV: A Christmas Carol)

Shortly after his regeneration, the Twelfth Doctor commented that he used to have a scarf when noting that he felt cold, but dismissed the memory as he considered the scarf as having looked ridiculous. (TV: Deep Breath)

The scarf eventually ended up in the Doctor's study on board the TARDIS, among other knick-knacks collected through his years of travel. (GAME: TARDIS, The Gunpowder Plot)

When the Tenth Doctor was traveling with Gabby Gonzalez, one of the scarves was on a mannequin with one of the Fourth Doctor's brown coats in the TARDIS wardrobe. (COMIC: Laundro-Room of Doom)

The Eleventh Doctor threw one of his scarves out of the way while in the TARDIS console room when he was searching for his sonic screwdriver. (COMIC: Sonic Sleuth)

Uses
The scarf proved to be useful on many occasions as something other than a piece of attire.

On the Nerva station, the Doctor unsuccessfully used it to try to deactivate the Auto guard. (TV: The Ark in Space)

It was used to measure a puzzle. (TV: Pyramids of Mars)

The Doctor escaped execution by using his scarf to trip up Count Federico's executioner. (TV: The Masque of Mandragora)

The scarf was used to trip Eldrad down a chasm, (TV: The Hand of Fear) and to trip the Master on the gantry. (TV: Logopolis)

The scarf briefly appeared in the Matrix on Gallifrey when it was used to prevent the Doctor from falling down a cliff face. However, the scarf disappeared not long afterwards. (TV: The Deadly Assassin)

The scarf was filled with small objects from the Doctor's pockets to try to provide a distraction for a predator resembling a giant bird louse that had chased the Doctor up a tree, but it was badly damaged before Leela arrived to save him. (PROSE: Last Man Running)

The scarf was used as a lead to drag K9 about when his power had run down. (TV: The Invisible Enemy)

The Doctor used the scarf to pull Romana I up a cliff face. (TV: The Stones of Blood)

The scarf was used as a rope to climb a mine shaft. (TV: The Creature from the Pit)

The Doctor's scarf had been wound about Stimson's corpse by his Foamasi killer in an attempt to frame the Doctor for the murder. (TV: The Leisure Hive)

Real world prop

 * According to Tom Baker, when a woman named Begonia Pope was asked to knit a scarf for the new Doctor, she was unsure how long a scarf was required. She consequently used all of the wool she had been given, resulting in a ridiculously long scarf. The producers loved it and, after it had been shortened slightly, used it for the Fourth Doctor's first story Robot.
 * There were many scarves used during Tom Baker's era. This included a considerably shorter "stunt scarf" for action sequences. The pattern and colouration varied from season to season, as did the overall length of the scarf, which, at one point reached a maximum length of 24 feet.
 * When June Hudson was asked to design a new outfit for the Doctor at the start of season 18, she was given the option to omit the scarf by producer John Nathan-Turner. Eventually, she decided the scarf had become too much a part of the Doctor's image to omit. She designed a new scarf to go with the burgundy colour scheme of the Doctor's new outfit.

Prime Computer ads

 * In 1980, Tom Baker played the Fourth Doctor in a series of TV commercials for Prime Computer. In one of the commercials, when asked by the Doctor of his scarf's length, Prime Computer responds, "It is 7.013 metres exclusive of the loose threads [referring to the tassels]!".

whoisdoctorwho.co.uk
The website whoisdoctorwho.co.uk had a list of sightings of the Doctor from which people had ostensibly been submitting to Clive, a conspiracy theorist character from Rose, who had pictures of the Doctor's ninth incarnation on the website, asking if anyone had seen him.

A submission from Dinah May claimed to have seen "this man several years ago, dumping some old clothes at the local tip", including a long scarf. When Dinah asked him about them, he claimed to have been "having a clear out". One of the items he was throwing away was a long scarf.