The Doctor's recorder

A recorder is a woodwind musical instrument played by blowing down a slit at the top and placing one's fingers and thumb on the airholes to produce notes.

The Second Doctor owned a recorder and often enjoyed playing it, usually as a source of self-amusement, but also to help him think, he used it on many occasions:

• He played it several times while on Vulcan. (DW: The Power of the Daleks)
• Played the recorder while in a Scottish prison cell. (DW: The Highlanders)
• The Doctor played the recorder while visiting Atlantis. (DW: The Underwater Menace)
• Played the recorder while visiting an Earth colony. (DW: The Macra Terror)
• Played the recorder while imprisoned by the Daleks. (DW: The Evil of the Daleks)
• Played the recorder while imprisoned in Tibet. (DW: The Abominable Snowmen)
• When the TARDIS landed in Australia, Jamie made the Doctor leave his recorder in the ship before they went out. (DW: The Enemy of the World)
• Played the recorder while under Robot Yeti guard. (DW: The Web of Fear)
• After escaping from the Land of Fiction, the Doctor realised he left his recorder there, and sent himself back in to retrieve it. While there, he encountered Goth, who asked him to help the Time Lords, and thus sent him to meet his future self. (DWM: Future Imperfect)
• Gave his recorder to Flora Millrace. (ST: The Avant Guardian)
• Used his recorder to control giant metal spiders on a jungle planet. (TVC: Master of Spiders)
• Used his recorder to befriend a herd of woolly mammoths on prehistoric Earth. (TVC: Pursued by the Trods)

Although his memories of meeting his future self quickly faded, the Doctor did remember that he had lost his recorder, and went to a music shop on Amber Station to get a new one. (DWM: Briefly Noted)

Other

When the Second and Third Doctor were transported to Omega's realm, the recorder fell into the TARDIS' force field generator, and was therefore not converted into anti-matter during the trip there. When the Doctors discovered this, they offered it to Omega, and the resulting annihilation seemingly destroyed the renegade Time Lord and his world. (DW: The Three Doctors)

Sarah Jane Smith discovered the Doctor's recorder in the TARDIS' secondary control room. (DW: The Masque of Mandragora)

The newly regenerated Doctor tried playing the recorder, but realised it was just not him. (DW: Castrovalva)

The Doctor played the recorder in an alien marketplace to amuse Lilith. (NA: Timewyrm: Apocalypse)

According to one account, the Ninth Doctor threw the recorder away on a rubbish tip on Earth while having a clear out of some old things. In this case, however, the instrument was referred to as a flute. (whoisdoctorwho.co.uk)