The Doctor's scarf

The Doctor's scarf was a scarf worn by the Doctor, predominantly by their fourth incarnation. At least two such scarves existed, one striped with a variety of colors (TV: Robot) and one that was primarily burgundy with purple stripes. (PROSE: Into the Silent Land) According to the Prime Computer, the first variant was "7.013 metres exclusive of the loose threads" in length, (TV: Step Into the 80's!) The scarf with the second pattern was later unravelled by the Fifth Doctor to leave a trail for himself to find his way back to the console room. (TV: Castrovalva) The Father of Time wore an identical scarf beneath his robes and held a recorder, two clues as to his true identity. (COMIC: The Test of Time)

Profile
Shortly after regenerating from his third incarnation, the Fourth Doctor accompanied Brigadier Lethbridge-Stewart on an investigation. He tried on several conspicuous costumes before settling on a Bohemian outfit. This included a wide-brimmed fedora and a long scarf. (TV: Robot) The scarf was originally knitted by Madame Nostradamus (TV: The Ark in Space), according to one source, as a gift for the First Doctor. (PROSE: "Unpredictable Tastes")

Upon regenerating, Romana II wore a copy of the Doctor's scarf (and the rest of his attire) after he told her to wear something more sensible and stylish. (TV: Destiny of the Daleks) According to one account this was actually a manifestation of the TARDIS disguising itself as Romana. (PROSE: The Lying Old Witch in the Wardrobe)

While listing pointless facts to defeat the Prolocture, the Doctor described five of the colours of his scarf as "red, green, chestnut, purple, and grey." (AUDIO: Babblesphere)

After Romana ordered the Doctor to clean his scarf, K9 Mark II once told him that after trailing across hundreds of worlds, the tassels of the scarf had grown a unique type of micro-culture developing into its own species. (PROSE: Doctor Who and the Krikkitmen)

Near the end of this incarnation, the Doctor switched to a longer scarf, but coloured in shades of burgundy. (PROSE: Into the Silent Land) 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, the Doctor used 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 occasionally tripped on it; his companion Erimem noted that it was so long she could practically mummify herself with it. (AUDIO: No Place Like Home)

The Seventh Doctor used the tartan scarf from his newly found outfit to stall as he tried to escape her laboratory. (TV: Time and the Rani) The Doctor replaced the lost scarf with a paisley one in the TARDIS wardrobe. (PROSE: The Useful Pile)

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

While selecting an outfit following his seventh regeneration, the Eighth 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)

Dinah May once saw the Ninth Doctor, some years prior to 2005, putting a flute, a velvet jacket, and a scarf into a local tip. When Dinah asked about the items, the Doctor told her he was just having a clear out. Dinah later wrote about this on the whoisdoctorwho.co.uk website in 2005. (PROSE: Have You Seen This Man?) The TARDIS wardrobe contained a multi-coloured scarf during the Doctor's tenth incarnation. (TV: The Christmas Invasion)

While the Tenth Doctor was travelling 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 and Kazran Sardick once wore scarves similar to the Fourth Doctor's when visiting Abigail Pettigrew. (TV: A Christmas Carol)

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

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)

Shortly after 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 TARDIS wardrobe still contained a very long multi-coloured scarf during the Doctor's thirteenth incarnation. Ryan Sinclair decided against wearing it deeming it a trip hazard. (PROSE: The Secret in Vault 13)

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

On the Nerva station, the Fourth Doctor made an unsuccessful attempt to use the scarf to deactivate the Auto guard. (TV: The Ark in Space) It was also 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)

Sarah Jane Smith also used the scarf as a rope to hold onto the Doctor when he tumbled off the edge of Scratchman's castle. She was not strong enough to pull him up, but the Cyber-Leader, who was given back his emotions by Scratchman, was. (PROSE: Scratchman)

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

The Doctor once got his long scarf caught in a pair of double doors at a UNIT Research Institute lab. While he exclaimed that he was caught in a force field, Sarah Jane Smith patiently worked to disentangle it. (PROSE: The Day of the Doctor)

The scarf 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)

When the Doctor and his Nest Cottage companions were absorbed by a Skishtari egg and stuck in a simulation based on the tale of Aladdin, the scarf became alive and took on the role of the genie of the tale. It was then able to provide an explanation for the situation and helped them escape by granting Andrew's wishes. (AUDIO: Aladdin Time)

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 lift 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)

Behind the scenes
In AUDIO: Aladdin Time, the scarf is voiced by Andrew Sachs.

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 and designed a new one to go with the burgundy colour scheme of the Doctor's new outfit.