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.
Series 3 rewrites with Rose seem popular.
It seems you don't actually get your Doctor Whofanfic writer's union card until you've written an "Eighth Doctor fights in the Time War and/or Ninth Doctor regenerates in the Time War and/or both angst about the Time War" fic.
More like Fandom Specific Title, but the Literary Allusion Title "World enough, and time" is practically irresistible to Whofic writers.
So you have a story. In it, the Doctor lands the TARDIS on Earth near where you live and meets a human with an inexplicable resemblance to you. He decides this human will be his new companion and the two of them go off to have wacky adventures together. Congratulations, we'll add it to the pile.
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.
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 the events of "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 is immortal because of Bad Wolf. Sometimes not because of Bad Wolf, oddly.
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 convices The Master to become good or The Doct decides to join forces with The Master and turn evil.
Ever since it was revealed his insanity was (partially) caused by the Time Lords implainting the drumming in his head, 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.