Caution: potential spoilers.

Doctor Who: Children in Nee (2005), on BBC One (about 8 minutes)
Council of Geeks; IMDb; TV Tropes; Wikia; Wikipedia; YouTube

This minisode (also known as "Born Again") aired on November 18, 2005, as part of the BBC's annual Children in Need telethon. It bridges the gap between the end of "Doctor Who" series 1 and the first Christmas special, The Christmas Invasion. The Doctor has just regenerated (he's now played by David Tennant), and he starts getting accustomed to his new body, while Rose has trouble believing he's really still the Doctor. It's actually quite dramatic, seeing the fear and confusion she's experiencing over her friend suddenly and unexpectedly being replaced by someone new, since she had no idea regeneration was a thing. It doesn't help that he starts acting manic, but he decides to take her home to London, on Christmas Eve. (I never saw the mini-special until 2015, on DVD, but it's not really necessary to see it, in order to understand what's going on where the Christmas special picks up.)

Attack of the Graske, on BBC Red Button
This aired December 25, 2005. It's some kind of interactive thing. I haven't seen it and don't plan on doing so.

Time Crash, on BBC One (about 8 minutes)
BBC (old page); Council of Geeks; IMDb; TV.com; TV Tropes; Wikia; Wikipedia

This aired on November 16, 2007, as part of the annual "Children in Need" telethon. It takes place between the end of series 3 and that year's Christmas special, Voyage of the Damned. I'm not at all sure it's canonical, but at least it's very amusing. Anyway, the tenth Doctor meets the fifth Doctor. I dunno what else to say, but you should be able to find it on YouTube or something, and I highly recommend checking it out for yourself.

Meanwhile in the TARDIS (home video)
These two minisodes were released on the DVD and Blu-ray sets of series 5, on November 8, 2010. I haven't seen them, and I don't know if or when I might get the chance.

Space / Time, on BBC One (about 3 to 4 minutes each)
Council of Geeks; IMDb; TV.com; TV Tropes; Wikia (Space); Wikia (Time); Wikipedia

These two minisodes, Space and Time, aired on March 18, 2011, as part of the BBC's biennial Comic Relief telethon. It takes place prior to series six. "Space" begins with the Doctor, and for some reason Rory, doing some sort of work on the tardis. But Amy accidentally ends up distracting Rory with her skirt, and... weirdness happens. That is to say, the tardis materializes inside itself. And then... a second Amy enters the tardis. And in "Time," things continue to get weirder, before finally the problem is resolved. But it's all bloody funny.

Death is the Only Answer, on BBC Three
This aired October 1, 2011, as part of Doctor Who Confidential. I haven't seen it and don't know if or when I might get the chance.

Night and the Doctor (home video)
This is a series of five minisodes that were released on the DVD and Blu-ray sets of series 6, on November 21, 2011. I haven't seen them and don't know if or when I might get the chance.

Good as Gold, on CBBC
This aired on May 24, 2012. I haven't seen it and don't know if or when I might get the chance.

The Great Detective, on BBC One
This aired on November 16, 2012, as part of the "Children in Need" telethon. It's one of three prequels to the Christmas special The Snowmen, though the other two were originally released online.

Rain Gods / Clara and the TARDIS / The Inforarium (home video)
These three minisodes were released on the DVD and Blu-ray sets of series 7, on September 24, 2013. I haven't seen them and don't know if or when I might get the chance.

Doctor Who: Children in Need (2023), on BBC One (about 5 minutes)
Council of Geeks; TV Tropes; Wikia; Wikipedia; YouTube

This minisode (also known as "Destination: Skaro") aired on November 17, 2023, as part of the BBC's annual Children in Need telethon. It bridges the gap between the end of "Doctor Who" 2022 specials and the 2023 specials. Davros is talking with someone named Castavillian about his new creation for the survival of the Kaled race, but he hasn't got a name for it yet. When Davros leaves the room for a couple of minutes, the Fourteenth Doctor (David Tennant) crashes his TARDIS into the room, damaging the machine, which he calls a Dalek, before realizing this was the very first of its kind, and wants Castavillian to pretend the Doctor was never there. I found it pretty amusing.

...I shall list more minisodes here whenever I get caught up on viewing subsequent series of Doctor Who.