A hatstand was a device used to store hats and coats. They hung from prongs upon the stand.
A hatstand stood in the hallway of Gabriel Chase, upon which the Seventh Doctor’s panama hat and question mark-handled umbrella hung for the duration of his and Ace’s time there. (Ghost Light)
The Doctor owned a hatstand and kept it in the console room of his TARDIS. The Fifth Doctor moved it out as he decided it no longer needed to have pride of place as he only wore a hat when he went out. The Doctor claimed to have another hatstand within the TARDIS and declared that he was to put them together so that he had a pair.
After the TARDIS was supposedly destroyed, the hatstand was all that was left. Turlough used it to threaten the colonists of Frontios, and they took it to be a device that called down meteorite storms. The Doctor and Turlough used this to their advantage, threatening the colonists into opening the research centre for them. The Doctor left the hatstand to the colonists as a gift upon his departure. (Frontios)
During his seventh incarnation, he brought a hatstand back into the console room, thinking the TARDIS needed a little something. (Daisy Chain)
Charley Pollard used the TARDIS hatstand to knock out Sebastian Grayle. (Seasons of Fear)
After the Eleventh Doctor flew the Pandorica into his exploding TARDIS, he came to himself in the TARDIS a week earlier by the hatstand. (The Big Bang)
In the Eleventh Doctor’s next TARDIS console room, the hatstand had disappeared, which The Doctor was very annoyed with when Clara Oswald asked where she could put her umbrella. (Hide)