Hostile Action Displacement System

The Hostile Action Displacement System, or HADS, was one of the defence mechanisms of the Doctor's TARDIS. When the outer shell of the vessel came under attack, the unit dematerialised the TARDIS and rematerialised it a short distance away after the attacker had gone, in a safer locale. (TV: The Krotons) However, another function of this was a "dispersal" instead of a "displacement", which would allow for the TARDIS to appear to actually be blown apart, but in fact remaining intact in various pieces. (TV: The Witch's Familiar)

The HADS had to be manually set, and the Doctor often forgot to do so. (PROSE: The Krotons)

When testing the ship after his exile on Earth had ended, the Third Doctor briefly lost the TARDIS when the HADS was triggered by a hailstorm, the Doctor noting to Jo Grant in embarrassment after they rediscovered the ship that the setting for the HADS was slightly too high. (PROSE: The Suns of Caresh)

While stuck on Earth and attempting to avoid meeting himself, the Fifth Doctor engaged the HADS to move the TARDIS. (AUDIO: The Mutant Phase)

The Fifth Doctor had General Tillington fire lasers at the partially-materialised TARDIS in order to trigger the HADS as part of his plan to defeat the Daleks. (AUDIO: Renaissance of the Daleks)

was able to capture the Doctor's TARDIS in a tractor beam because the Sixth Doctor had neglected to set the HADS. (PROSE: Time and the Rani)

Ace activated the HADS on instructions from the Seventh Doctor, moving the TARDIS from Haiti's Presidential Palace to the university. (PROSE: White Darkness)

The HADS activated itself when two Zygons tried to steal the TARDIS. (PROSE: The Bodysnatchers)

The HADS activated again when temporal corrosion threatened the TARDIS, sending Charley Pollard from 1 January 2008 to 15 January 1942. (AUDIO: The Girl Who Never Was)

After a particularly violent crash landing, the TARDIS used the HADS to move itself through time as well as space, appearing several days later. (PROSE: Dead of Winter)

The Tenth Doctor tried to activate the HADS after Finn Dargo had hacked Emergency Program One and attempted to arrest the TARDIS' occupants. The move allowed the TARDIS to materialise in an unexpected place in Dargo's ship, giving Majenta Pryce a slight advantage over Dargo's Intersol team. (COMIC: The Crimson Hand)

While fiddling with various settings, the Eleventh Doctor re-enabled the HADS in time for it to activate and leave the Doctor and Clara Oswald in a sinking submarine near the North Pole in 1983. It reappeared a short while later, but at the South Pole instead of the North Pole. (TV: Cold War) This event would later be referenced by Clara when the Twelfth Doctor assured her that the TARDIS would "turn up somewhere" after the space shuttle it had materialised in fell into a canyon. (TV: Kill the Moon)

When the Daleks attempted to destroy the TARDIS, (TV: The Magician's Apprentice) the HADS activated making it look like it had been destroyed, but the Twelfth Doctor later rematerialised the TARDIS using his sonic sunglasses, summoning pieces of it from all around the room; this action was called the "Hostile Action Dispersal System". (TV: The Witch's Familiar)

Behind the scenes
According to the Make Your Own Adventure with Doctor Who novel, Search for the Doctor, Drax's TARDIS was shown to have a HADS, which was manually triggered when the vessel came under fire. As the Make Your Own Adventure series has multiple outcomes and no mechanism of determining the "right" outcome, this wiki does not consider the novel a valid source.