The Doctor's scarf

Although throughout his lives, the Doctor owned many scarves, the one best remembered is the Doctor's scarf from his fourth incarnation. At least two such scarves existed. (DW: Robot, The Leisure Hive)

Profile
Shortly after his third regeneration, the Doctor 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 knitted for him 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: The 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 the Doctor unravelled his burgundy scarf in the the TARDIS' corridors, allowing his companions to follow him. (DW: Castrovalva)

The 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 original scarf in order to hide her non-human body. (DWM: The Way of All Flesh)

The Doctor suposedly 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 [sic], 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 Doctor and Kazran Sardick were both seen to wear scarves similar to this one 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 Robot.
 * There were many scarves used during Tomb Baker's era. This included a considerably 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 Producer John Nathan-Turner. Eventually, 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 new outfit.