Doctor Who/Shout Out

In "The Robots of Death", "Grimwade's Syndrome", a fear of robots, was named after director (and, later, writer) Peter Grimwade, who hated to do robot stories. The same story had a character named Poul, presumably after Speculative Fiction writer Poul Anderson. Given the robot theme, Uvanov in the same story may have been a slightly subtler reference to Isaac Asimov, and Taren Capel to Karel Capek. In the novel Corpse Maker, a sequel to "Robots of Death" by the same writer, Poul is given the first name Andor.
In "Blink", policeman Billy Shipton tells Sally Sparrow the TARDIS can't be a real police box because "the windows are too big" (amongst other things). It's been confirmed by episode writer Steven Moffat that this was a reference to complaints made by fans on the popular Outpost Gallifrey discussion forums in 2004. (Likewise "Time Crash" and "Love & Monsters" both make oblique reference to Fandom.)
In "Time Crash", the Fifth Doctor makes a Precision F-Strike, or rather a Precision B-Strike.
"The Shakespeare Code": New companion Martha remarks that the apparent witchcraft being done "all sounds a bit Harry Potter", The Doctor enjoins her to wait till she reads the 7th book (which hadn't come out when the episode aired). Later, they and Shakespeare have to, on the fly, say the right words to banish the Carrionites:

Shakespeare: Banished like a tinker's cuss, I say to thee... (looks to the Doctor)
The Doctor: Uhh... (looks to Martha)
Martha: Expelliarmus!
The Doctor: Expelliarmus!
Shakespeare: Expelliarmus! (It works)
The Doctor: Good old J.K.!

Some Captain Scarlet fans were quick to note that the Valiant looked a lot like Cloudbase.
In the 1988 serial "Dragonfire", the Doctor and a philosophically minded guard debate the concept that "the semiotic thickness of a performed text varies according to the redundancy of its auxiliary performance codes". This is a direct quote from an academic media studies book entitled Doctor Who: The Unfolding Text, and roughly translates into laymen's terms as "the less relevant an in-joke is to the plot, the more cultural significance it has".
In "Human Nature", the Doctor's human alter-ego gives his parents' names as Sydney and Verity. Sydney Newman was one of the original creators of the show, and Verity Lambert was its first producer... they could very well be considered the "mother and father" of Doctor Who! This is reprised in "The End of Time", the 2009 Christmas special, with a cameo appearance by one of Joan Redfern's descendants, who has written a book based on John Smith's journal under the name "Verity Newman".
"Partners in Crime" contains what may well be a shout to us, since Donna uses the term "planet of the hats". Some fans argue that the same episode contains a reference to one of the people who regularly bittorrents new episodes for those of us on the other side of the Atlantic.
"Forest of the Dead" has Donna's two "children" called Josh and Ella. Josh is the name of Steven Moffat's son, with Ella being a friend of his.
The 2005 Doctor Who webcast "Shada", based on a script by Douglas Adams, included a Nutrimat drinks machine and a Ford Prefect car, both references to The Hitchhiker's Guide to the Galaxy.
A similar reference was made in The Christmas Invasion: "Not bad for a man in his jim-jams. Very Arthur Dent. Now, there was a nice man."
- In the same episode, the Doctor entreats the Sycorax to leave the humans alone. He makes a heartwarming speech about how from the day they arive on the planet, and blinking, step into the sun, there is more to be seen than can ever be seen... whereupon he remarks, "Wait, that's The Lion King."
Way back in "Ghost Light", the Doctor asked "Who was it who said Earthmen never invited their ancestors to dinner?"
The 2007 episode "42" plays out in real time, à la 24.
- The title, of course, is another Hitchhiker's reference.
"The Unicorn and the Wasp": "I've gathered you all here in the Accusing Parlour..."
- To say nothing of the number of Agatha Christie titles dropped into the dialogue ("Well, that's put The Cat Among The Pigeons", "It's a trick, an illusion, They Do It With Mirrors", "The Moving Finger points to...")
  - Word of God is that the episode was inspired by the board game Clue ("Cluedo" in the UK).
"Army of Ghosts" sees the Tenth Doctor donning a "triangulation kit" consisting of a power-pack and three TARDIS roundels; the finished getup resembles the Ghostbusters' infamous Proton Pack. Cue a brief rendition of Ray Parker, Jr's classic theme tune.
During "The Beast Below" Amy runs around in her nightie and a dressing gown the whole time, a bit like a female version of Arthur Dent.
Of course, The Cybermen and Borg have obvious similarities, but they made it even more obvious when a Cyberman said "YOU WILL BE ASSIMILATED!" in the episode "The Pandorica Opens."
- And the dart it shot just before that is definitely of Kaminoan design, complete with poison (only not lethal). Contains Fridge Brilliance, as Amy has never met the Cybermen before that point, so in the scenario based on her own memories the introduction is made to maintain her immersion by means of Shout Outs to known sci-fi franchises.
The sequence in The Time of Angels with the Weeping Angel slowly emerging from the television screen in is quite possibly a reference to The Ring.
The Doctor's Only Mostly Dead quote in The Big Bang is a clear homage to The Princess Bride.
In the special The Waters of Mars, a character is infected by a single drop falling onto his eye. Remind you of anything?
In "The God Complex", the Doctor explains the configuration of a prison, with multiple rooms changing shape and form and new threats to face each time. Then he flicks up a Rubik's cube and comments "Well that's just rude".
The Empty Child and The Doctor Dances seem to contain a shoutout to The Importance of Being Earnest with Jack shagging Algy, with Jack and Algy (short for Algernon) being main characters in The Importance of Being Earnest.