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, DW: The Leisure Hive) The second was later unraveled by the Fifth Doctor to leave a trail for himself to find his way back to the console room. (DW: 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. (DW: Robot) The scarf had originally been knitted for him by Madame Nostradamus. (DW: 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. (DW: The Leisure Hive) However, his original scarf was 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. After this, he began using pieces of his former self's clothing to continue the trail before finding a cricketeer outfit. (DW: Castrovalva)

The Fifth Doctor admitted that the scarf was warm and sometimes useful, 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, though he did think about using it for a moment. (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 to hide her non-human body. (DWM: The Way of All Flesh)

The Ninth Doctor supposedly 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 contained a multi-colour scarf during the Doctor's tenth incarnation. (DW: The Christmas Invasion)

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

The scarf eventually ended up in the Doctor's study onboard the TARDIS, among other knick-knacks collected through his years of travel. (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)
 * The scarf was used to trip Eldrad down a chasm (DW: The Hand of Fear), and to trip the Master on the gantry. (DW: Logopolis)
 * The scarf was used in the Matrix on Gallifrey to prevent the Doctor from falling down a cliff face. (DW: The Deadly Assassin)
 * The scarf was used as a lead to drag K9 about when his power had run down. (DW: The Invisible Enemy)
 * The scarf was used as a rope to climb a mine shaft. (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 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 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. She designed a new scarf to go with the burgundy colour scheme of the Doctor's new outfit.
 * Tom Baker reprised his role of 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!".