The episode "Doomsday" spawned hundreds, if not thousands, of 10/Rose Reunion fics, an event which became canon with "Journey's End". After that, many are writing 10.5-and-Rose-in-the-alternate-universe fic.
Despite this there are many fans who can't accept the splitting and will try any means to get the Doctor and Rose together. Expect from both a lot ofFan Dumb.
Stories in which Donna recovers her memories are quite common, usually taking place either at the end of her life (usually elderly) with the Doctor visiting her one last time, or by miraculously being able to remember everything without her mind burning itself up in the process. The latter is accomplished by the Doctor figuring out some clever way to save her that he didn't before, or by regeneration.
The "Donna recovers her memories and also becomes a Time Lord" sub-genre is also quite popular.
Fics where the Doctor regenerates into an eight year old/an animal/a girl/pretty much anything else.
River Song's true identity used to be a mystery and many different scenarios and theories were written about it, usually by somehow making her be a future version of a companion, namely Rose Tyler, Donna Temple-Noble or Amy Pond. While they significantly died down since her identity was revealed in "A Good Man Goes To War" and "Let's Kill Hitler", Alternate Universe Fics are still being written that ignore some or all the events of those episodes.
Also relating to River, "Forest Of The Dead" Fix Fics are common, as well as fanfics involving her and the Doctor going to the Singing Towers.
There are practically just as many multi!Doctor crossover stories as there are single Doctor ones.
Rose becomes immortal, usually because of Bad Wolf.
Due to the sheer amount of Foe Yay in the series, there are tons of fanfics centered around The Master and The Doctor having a romantic relationship with one another. Bonus points if either The Doctor convinces The Master to become good or The Doctor decides to join forces with The Master and turn evil.
Ever since the Master's backstory was revealed in series 3, angsty Master stories have become more popular.
Likewise, stories about The Master trying to redeem himself have become popular as well.
Intra-fandom crossovers involving multiple TARDIS teams meeting, often out of order, have become increasingly popular on sites such as Teaspoon.
During "The Year That Never Was", the Master rapes the Doctor, Jack, and/or Martha, with a heavy bias toward the Doctor. Note that he (almost?) invariably reverts the Doctor to his usual physical age before doing so. Usually played as erotic yet angsty.
Female Doctor fics are a common theme in Doctor Who fics. Expect FemDoctor to be paired with either Jack or The Master.
Post-"The Angels Take Manhattan", a large number of fics tend to focus on the Ponds adjusting to life in 1940s Manhattan. And, as always, we have the Fix Fics where River brings them back to the present.
The Eighth Doctor, in the one TV episode that we saw him in, was presented as a rather flighty, optimistic, romantic and sensitive sort. Naturally, this practically makes him a magnet for Hurt/Comfort Fic. While the new series Doctors get plenty of this themselves seeing as they practically have "tortured soul who's been put through too much and needs lots of big hugs" written into their DNA thanks to the Time War, something about the Eighth Doctor practically demands the author to do horrible, horrible things to him.
Rory's 2000 years guarding the Pandorica, and his unsuccessful attempts to stay out of trouble, are a major fanfic magnet.
The Doctor spending hundreds of years protecting Trenzalore from the various monsters that seek to destroy it.
Following the death of Elisabeth Sladen, fics featuring The Doctor learning about Sarah Jane Smith's death and coming to terms with it became common. Sometimes, they would have multiple companion cameos, and some would be Multi-Doctor fics. (Canonically, Sarah Jane is still alive, but the precedent of the Brigadier dying after the death of his actor makes such stories logically hard to resist.)