Blue

Blue was a colour between indigo and green on the visible spectrum. (TV: Sky)

As noted by the Sixth Doctor, it was the official colour of mourning on "a number of civilised worlds" including Necros, but notably not Earth. As such, he wore a blue coat as a mourning suit. (TV: Revelation of the Daleks, WC: Real Time/AUDIO: Real Time)

It was also associated with the Doctor's TARDIS, (TV: The Big Bang) as this was the colour of the police box disguise of the TARDIS. The TARDIS was often referred to as a "(big) blue box", even by the Doctor. (TV: Planet of Fire, Aliens of London, The Satan Pit, The Fires of Pompeii, Utopia, Death of the Doctor, The Doctor's Wife, Time Heist, Last Christmas, et al.) When the Eleventh Doctor sent a blue envelope to Amy Pond and Rory Williams, Amy knew he was the one who sent it, recognising the colour as "TARDIS blue". (TV: The Impossible Astronaut)

Blue could also be an eye colour. Many of the Doctor’s incarnations possessed blue eyes, more specifically the First, Second, Third, Fourth, Fifth, Sixth, Seventh, Eighth, Ninth and Twelfth Doctors (TV: The Three Doctors, Spearhead from Space, Robot, Logopolis, The Caves of Androzani, Time and the Rani, Doctor Who, Rose, The Time of the Doctor). The eyepiece and vision of a Dalek eyestalk were most often blue (TV: Dalek, The Parting of the Ways, Doomsday, Evolution of the Daleks, The Stolen Earth, et al).

Some incarnations of the Master possessed blue eyes, more specifically, his original incarnation as a child, (TV: The Sound of Drums) his, (TV: The Keeper of Traken) his upon escaping the Time War, (TV: Utopia) and the one who called herself. (TV: Deep Breath)

The sixth incarnation of the Corsair had blue eyes. (COMIC: Old Friends)

Adolf Hitler had blue eyes. (PROSE: Timewyrm: Exodus)

Romana I once tried on a blue-skinned body during her first regeneration, but soon decided against it. (TV: Destiny of the Daleks)

The Tenth Doctor sometimes wore a blue blazer. (TV: Smith and Jones, etc.) It was also worn by Corin. (TV: Journey's End, etc)

The Seventh Doctor wore a plain blue paisley necktie. (PROSE: Strange England)

During her trip to New Earth with the Tenth Doctor, Rose Tyler wore a blue zip-up jacket, which she swiftly removed when she was possessed by the Lady Cassandra. (TV: New Earth)

Many silver Daleks had blue sense globes. (TV: The Daleks, etc) Some of the Humanised Daleks who lived in Alpha's colony on the planet Kyrol following the Dalek Civil War were Blue Daleks. (COMIC: Children of the Revolution) The Blue Daleks were the equivalent of sergeants and corporals. (PROSE: War of the Daleks) The Dalek Time Controller had a blue casing. (AUDIO: To the Death, X and the Daleks) During the Last Great Time War, the Dalek members of the Eternity Circle were mostly blue, with silver domes and globes. Other Daleks present at the Daleks' temporary base in the city of Andor on Moldox were silver with blue domes. (PROSE: Engines of War) During the war, the Dalek Scientific Division were bronze with blue domes and sense globes. (AUDIO: A Thing of Guile)

In Through the Eye of Eternity, the Blue Eye of Eternity generally symbolised that the week's episode would be a spy drama. This was a reference to the "Cold" War. (PROSE: The Book of the War)

Metebelis III was the famous blue planet of the Acteon Galaxy. (TV: Carnival of Monsters, The Green Death) On the Blue Planet, almost everything was blue, including the natives. (PROSE: Dr. Sixth)

The sun briefly turned blue when the Slitheen began draining its power. (TV: Revenge of the Slitheen)

Blue Daleks and Strategist Daleks were identified by their blue casings. (PROSE: War of the Daleks, TV: Victory of the Daleks) Humanoid Daleks, (COMIC: Genesis of Evil) Vogans, (COMIC: The Vogan Slaves) Monstrons, (COMIC: The Menace of the Monstrons) Fodorans, (PROSE: Kursaal) Crespallions (TV: The End of the World), and Zerovians (COMIC: Impasse) had blue skin. Dahh-Ren was blue, (TV: Oxygen) as was Jorj. (TV: World Enough and Time) Arun's skin had a blue tint after the Panjistri experimented on her. (PROSE: Timewyrm: Apocalypse)

Nardole recalled being blue at some point, and suggested that he should "go back" to it. (TV: World Enough and Time)

The ears of a Fostarone turned blue when they lied. (PROSE: Ghosts of India)

In the spring of 2007, while facing a Hoix with the Tenth Doctor, Rose Tyler threw a blue bucket of liquid at the Hoix. This was the wrong bucket, Rose had thought the Doctor said blue, but as the Doctor noted, he had said "not blue". This enraged the Hoix who chased Rose and the Doctor until Rose found the correct red bucket. (TV: Love & Monsters)

Dan Lewis' horoscope for 2021 foretold that the colour blue would be important, among other things. (TV: Welcome to the TARDIS...)

While searching for a distress signal at the 1966 World Cup Final, the Fourteenth Doctor noticed blue fringing emanating from a group of four time tourists as a result of his sonic screwdriver interacting with their psychic shields. (COMIC: Liberation of the Daleks)

Other references
"Out of the blue" was a saying used both by Tish Jones (TV: The Sound of Drums) and her sister Martha Jones. (TV: Reset)