The Doctor's scarf

The Doctor's scarf was a long, multicoloured piece of knitwear that was an important part of the Doctor's outfit during his fourth incarnation. At least two such scarves existed. (DW: Robot, Leisure Hive)

When the Doctor, shortly after his third regeneration, had to accompany the Brigadier on a journey, he tried on several inappropriate costumes before settling on a vaguely bohemian outfit, which included a wide brimmed hat and a long scarf. (DW: Robot) The scarf had originally been made for the Doctor by Madame Nostradamus. (DW: The Ark in Space)

The Doctor once mentioned that, had he not produced a ball of string for Theseus and Ariadne, they would have unravelled his scarf to escape the labyrinth. (DW: The Creature from the Pit)

Later in this incarnation, the Doctor switched to a scarf of similar length, but coloured in shades of burgundy. (DW: Leisure Hive) However, his original scarf was seen to be kept on a hatstand in the console room. (DW: Warriors' Gate)

After his fourth regeneration, the Fifth Doctor unravelled his burgundy scarf in the TARDIS corridors, allowing his companions to follow him. (DW: Castrovalva)

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

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. (DW: Doctor Who)

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

The Ninth Doctor supossedly sold one of his scarves, as Dinah May posted in whoisdoctorwho:


 * I saw this man several years ago, dumping some old clothes at the local tip. There was a flute, a velvet jacket and a long scarf. I asked him about them, and he just said he was having a clear out. Then he said he had never really liked cricket either. 

The TARDIS wardrobe was seen to contain a multi-colour scarf during the Doctor's tenth incarnation. (DW: The Christmas Invasion)

The Eleventh Doctor and Kazran Sardick were both seen to wear scarves similar to the Fourth Doctor's during their earlier adventures when visiting Abigail Pettigrew. (DW: A Christmas Carol)

The scarf eventually ended up in the Doctor's study onboard the TARDIS. (VG: TARDIS, The Gunpowder Plot)

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. (DW: The Ark in Space)
 * It was used to measure a puzzle. (DW: Pyramids of Mars)
 * The Doctor escaped execution by using his scarf to trip up Count Federico's executioner. (DW: The Masque of Mandragora)
 * Tripping Eldrad down a chasm. (DW: The Hand of Fear)
 * In the Matrix on Gallifrey to prevent the Doctor from falling down a cliff face. (DW: The Deadly Assassin)
 * As a lead to drag K9 about when his power had run down. (DW: The Invisible Enemy)
 * Climbing a mine shaft using the scarf as a rope (DW: 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. (DW: The Leisure Hive)

Behind the scenes

 * 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.
 * There were many scarves used during the Tom Baker era, including a much shorter "stunt scarf" for action sequences. The pattern and coloration varied from season to season, as did the overall length of the scarf, which, at one point reached a maximum length of 24ft (nearly four times the height of Tom Baker himself!).
 * 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 John Nathan-Turner. In the end, she decided the scarf had become too much a part of the Doctor's image to omit, and designed a new scarf to go along with the new Burgundy colour scheme of the Doctor's outfit.
 * In VG: TARDIS, the Fourth Doctor's scarf can be seen in the drawing room.