The following list is very long and likely to get much longer, as Steven Moffat has openly admitted that he's having "endless fun" with finding modern parallels for aspects of the original stories.
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A Study in Pink
John is instructed to keep a blog, which he doesn't actually use until he signs up with Sherlock, at which point he records their mystery solving. It's very popular. Traditionally Watson publishes books about his and Holmes's mystery solving, and those were also rather popular.
One of the victims was a guy walking along in a downpour, who told his mate to go on without him while he popped back for his brolly. In The Problem of Thor Bridge, Watson mentions some of the cases he hasn't written up yet, one of which "...is that of Mr. James Phillimore, who, stepping back into his own house to get his umbrella, was never more seen in this world".
Sherlock whipping the corpse in his first scene is a shout out to a reference in A Study In Scarlet in which Stamford tells Watson that he saw him "beating the subjects in the dissecting-rooms with a stick."
There's a brief shot in 221b Baker Street of Sherlock pinning some papers to the mantelpiece with a small knife. In the books this is exactly how Holmes keeps his unanswered correspondences.
In the very first episode, Sherlock exclaims, "the game, Ms. Hudson, is on!" Doubtlessly a modernized version of "the game's afoot!", no?
Sherlock's deductions from John's phone are based on his deductions from Watson's pocket watch in The Sign Of Four.
In the Sherlock Holmes story A Study In Scarlet, the police assumed that the person writing Rache was trying to write the name "Rachel", and Holmes poo-pooed this by advising them that "Rache" is German for "revenge". However, this is completely inverted in "A Study in Pink".
Watson has no luck at all getting the attention of Mycroft's female aide. In the books, Watson was reputed as something of a ladies man.
John finds Sherlock lying down on the sofa moving his arm in a way that made it look (from the audience's POV) as if he had just treated himself to a ringer of cocaine.
It's a three-patch problem. = It's a three-pipe problem.
Both versions of "A Study in Pink" have John's phone being used to text the dead woman's phone because Sherlock's number is on his website and might be recognized. A Study in Scarlet had a newspaper advertisment in Watson's name because Holmes had gained some minor credit in the press from previous cases and so his name might have been recognized.
A character who is a Red Herring for being the murderer is an American visiting London for the first time. This is a nod to the original A Study in Scarlet, in which the murderer was an American visiting London for the first time. In both, the killer is a cabbie.
When Sherlock is standing, the shock blanket looks suspiciously like his Iconic Outfit.
Mycroft is the British government, according to Sherlock.
John reveals that he was actually shot in the shoulder, not the leg, referencing Doyle's habit of forgetting where exactly his wound was in the books.
In both cases he was shot in a recent Afghanistan War, though ...
Mrs. Hudson says several times in the first episode "I'm not your housekeeper, you know..." Indeed she isn't, but many people miss that the books refer to her as Holmes's landlady.
On John's blog, he reveals that Sherlock apparently doesn't know the Earth goes around the Sun, as mentioned in A Study in Scarlet. This is also referenced in episode 3.
Sherlock's explanation for why he doesn't memorize "useless" facts is updated from his head being an "attic" with limited space to a "hard drive".
The Blind Banker
The book code in "The Blind Banker" is similar to the one in The Valley of Fear.
Another nod to "The Valley of Fear" is the concept of someone hiding at his own home/workplace
There are also countless references to "The Dancing Men". Starting with a love story between an English man and a woman with a shady past, over the concept of a criminal leaving messages disguised as scribbling, the woman in question being threatened by such a message by a man from her past to a murder which looks like suicide because the bullet went through the open window.
A killer with small feet who can climb is also featured in "The Sign of the Four"
It has been established that Holmes uses the drawing room for target practice with a pistol. In the books, with "VR" for "Victoria Regina". The modern equivalent would be "EIIR" (for "Elizabeth (The Second) Regina"), which does SORT OF look like a smiley face, if you turn you head and squint
"The Great Game" is full of these. The scene with the stationery is "A Scandal in Bohemia", the pips are a sly nod to "The Five Orange Pips", the Bruce-Partington plans are a subplot of sorts...
"Domestic bliss must suit you Molly, you've put on another three pounds since I last saw you." "Two and a half." "No, three." From a Scandal in Bohemia (though it's Watson, and seven and a half pounds).
Sherlock repeatedly refers to the Czech Republic by the anachronistic name "Bohemia", just to make sure we get the reference.
"Any ideas?" "Seven, so far..." From The Adventure of the Copper Beeches.
Sherlock's network of homeless people is a deliberate update to the Baker Street Irregulars, a gang of street kids Holmes would pay to spy on people for him.
"I'd be lost without my blogger" is from Scandal again, where it's "Boswell".
"You do see, you just don't observe!" Also from Scandal in Bohemia, but delivered to Lestrade instead of Watson this time around.
Sherlock makes several digs about his brother's weight. Mycroft is traditionally portrayed as extremely fat and sedentary, though Gatiss is surprisingly trim for the role.
Moriarty's line to Sherlock near the end of "The Great Game": "I mean, I'm gonna kill you anyway, someday..."
Large chunks of the dialogue are from the one meeting in "The Final Problem", including Moriarty asking what Sherlock has in his pocket.
The hilarious stance Sherlock briefly adopts during his fight with the Golem might be a reference to the Victorian Sherlock Holmes' boxing skills. Or, a Shout-Out to Jeremy Brett's version, as Brett would adopt the same pose in fight scenes.
Sherlock and Jim's final exchange in "The Great Game" exactly mirrors the opening of their first exchange in "The Final Problem":
Moriarty: All that I have to say has already crossed your mind. Holmes: Then possibly my answer has crossed yours.
In "The Great Game", Sherlock asks John to make deductions about the pair of shoes, and then congratulates him before saying he missed out almost everything of importance. He does the same in A Case of Identity.
The title itself is a reference too, since fans refer hunting down canon clues and trying to find an explanation for conflicting information as "The Great Game". Plus, Holmes himself refers to his work as "playing the Game" multiple times.
A Scandal in Belgravia
The early minutes of "A Scandal in Belgravia" go through a proper Hurricane of Puns over the names of many of the original novels: The Geek Interpreter, The Speckled Blonde, The Navel Treatment...
Sherlock's analysis of tobacco ash on his website - in "The Boscombe Valley Mystery," Holmes remarks that he has "written a little monograph on the ashes of 140 different varieties of pipe, cigar, and cigarette tobacco".
Sherlock throws on a deerstalker cap to try to disguise himself from a throng of new-found fans. And much like his real-world fans, these fans get the impression that he wears it all the time.
"This isn't a deerstalker anymore, it's a Sherlock Holmes hat."
In that scene Watson also grabs a hat: a flat cap similar to the one worn by Nigel Bruce (The Watson to Basil Rathbone's Holmes).
This conversation at Christmas, with Sherlock obsessing over John's blog:
Sherlock: You've got a photograph of me wearing that hat?!
The severed thumbs in the fridge are a nod to "The Engineer's Thumb".
The bit of dialogue where Sherlock asks "And my client is?" and the government official answers "Illustrious" is another reference to a story title, "The Illustrious Client." Sherlock's remark that "I'm used to mystery at one end of my cases, both is too much work," is lifted from that story as well.
At Buckingham Palace, the government official tells John that his employer likes his blog, "particularly the one about the aluminium crutch". In the stories, one of Holmes' early cases before he met Watson revolved around an aluminium crutch.
Irene Adler's most common alias is given as "The Woman." The first line of "A Scandal in Bohemia" is "To Sherlock Holmes she is always THE woman."
Many of the details of Sherlock and John's first meeting with Irene Adler (the priest disguise, the fire alarm ruse, "Goodnight Mr. Sherlock Holmes") are the same as they were in the original story.
John's suggestion of "Hamish" to Sherlock and Irene ("John Hamish Watson. Just, if you're looking for baby names.") is as much a Fandom Shout-Out as a Mythology Gag. In the stories, Watson's name is almost always given as "John H. Watson" — but in one, his wife addresses him as "James", not "John". William S. Baring-Gould concluded that the "H" must stand for Hamish, the Scottish form of "James", and that Mrs. Watson uses this as a pet name for her husband; this has since been adopted as Fanon.
The ending bears a strong resemblance to the film The Private Life of Sherlock Holmes in that Holmes is tricked by a female spy into accidentally exposing one of Mycroft's top-secret government projects. He is then told by Mycroft that said spy has been sent away safely overseas, only to find out later that she's been executed. Although, in Belgravia, it turns out she's Not Quite Dead.
The Hounds of Baskerville
Sherlock harpooning a pig carcass to solve a case comes straight from "Black Peter".
Sherlock's frustrated "I need something stronger than tea! Something 7% stronger..." is a reference to his famous cocaine habit in The Sign of Four, where he injected a 7% solution.
When Sherlock is having his nicotine freak out, one of the places he looks for his hidden stash is in the toe of an old Persian slipper.
Sherlock's remark about his mind being "like an engine racing out of control, a rocket tearing itself to pieces on the launch pad" when he doesn't have work is very similar to a line from "The Adventure of Wisteria Lodge".
Henry Knight's name is a play on "Sir Henry" from The Hound Of The Baskervilles. Many of the other names (Mortimer, Frankland, Stapleton, Lyons, Barrymore, Selden) are taken more or less straight from the original - including the surname of the original villain. Interestingly, the character with the original villain's name turns out to be a Red Herring who eventually helps Sherlock and John to solve the case, but the real villain of this story did actually have similar personality traits to the original and also dies in a very similar way.
Sherlock's waffling about whether or not he will go to Dartmoor or send John to investigate is a wink at the fact that, in the Conan Doyle version, Holmes does send Watson down to Baskerville Hall in his place. Sort of.
One of the innkeepers mentions he has trouble sleeping at night due to "the ruddy prisoner". In the original novel, an escaped killer was on the loose in the area.
The Grimpen Minefield is, of course, wordplay on the Grimpen Mire. The villains in both stories die trying to flee through it.
Outside the pub, Sherlock spots the horse-racing section of a paper rolled up in the back pocket of a potential eyewitness. When the man is not forthcoming with his information, he pretends to have bet John that the man couldn't produce proof, and the man immediately gets very talkative. The original Holmes pulls the exact same stunt in "The Blue Carbuncle".
The tour guide who claims to have seen the hound is named Fletcher - presumably in reference to Bertram Fletcher Robinson, who originally gave Conan Doyle the idea for a ghostly hound.
Major Barrymore sports a thick dark beard, in spite of the fact that it's against army regulations. This is because the original character's most distinguishing physical characteristic was a black beard.
Mycroft is seen briefly sitting alone in a place that seems awfully similar to The Diogenes Club. It is later established that the Diogenes Club does in fact exist in the show's universe.
John's incredulity over Sherlock's "plan" (take Henry out on the moor and see if anything attacks him) is particularly funny considering that Holmes' master plan in the novel also basically boils down to using poor Sir Henry as bait.
"Once you've ruled out the impossible whatever remains, however improbable, must be true," is a favorite Holmes Catch Phrase employed in several of the original stories.
Sherlock's remark that emotion is like "grit on the lense, a fly in the ointment," is a reference to the line "Grit in a sensitive instrument, or a crack in one of his own high-power lenses, would not be more disturbing than a strong emotion in a nature such as his" (From "A Scandal in Bohemia").
The flashing lights on the moor that John takes for Morse code signals are a nod to the escaped prisoner subplot of the novel. When Watson goes to investigate, the man having sex in the car is referred to as "Mr. Selden." Selden was the prisoner's name.
In addition to the major plot points and most of the character names, "The Hounds of Baskerville" keeps arguably the two most famous quotes from the original: "Mr. Holmes, they were the footprints of a gigantic hound!" and Holmes' reference to Watson as a conductor of light. "Murder - refined, cold-blooded murder" is also lifted straight from the novel.
Also "Stay away from the moor if you value your life."
Sherlock describes Lestrade as being "brown as a nut" (meaning his tan). This is lifted word-for-word from "A Study in Scarlet," although there it was Stamford describing Watson.
The glowing bunnies are a nod to the original hound, which was painted with phosphorus to appear supernatural.
The hallucinogenic fear gas is reminiscent of "The Devil's Foot," as is Sherlock's experiment with it, although in this case the results are merely emotionally traumatizing rather than life-threatening.
Sherlock's experiment on Watson also recalls a remark of Stamford's in "A Study in Scarlet" that he could imagine Holmes "giving a friend a little pinch of the latest vegetable alkaloid, not out of malevolence, you understand, but simply out of a spirit of inquiry...".
The Reichenbach Fall
Moriarty refers to his conflict with Sherlock as "The Final Problem", referencing the name of the short story this episode is based on. Likewise, Reichenbach Falls made a prominent appearance in that story as the location where Holmes (apparently) tumbles over the ledge to his doom.
Peter Ricoletti, the man at the top of Interpol's most-wanted list, is a reference to one of Holmes' untold early cases ("Ricoletti of the club foot") mentioned in "The Musgrave Ritual."
Sherlock's deduction about Kitty Reilly's wrist (pressure marks from resting it on the desk while typing) is similar to one that Holmes makes in "A Case of Identity."
Sherlock's courtroom description of Moriarty as a spider at the center of a criminal web is lifted almost word-for-word from "The Final Problem."
Moriarty's threatening but very civil discussion with Sherlock in the sitting room at Baker Street recalls his visit to 221B in "The Final Problem."
During their tea-party, Moriarty tells Sherlock, "You need me - or you're nothing." Holmes himself expresses a similar sentiment in "The Adventure of the Norwood Builder."
The Diogenes Club is from the original stories, as is its blanket ban on talking.
The abduction of the diplomat's children recalls "The Adventure of the Priory School," in which Holmes is called in to find the young son of a prominent nobleman, who has also been kidnapped from a high-class boarding school.
As in the original "Final Problem," Sherlock is nearly run down by a cab. The twist is that it's not an assassination attempt by Moriarty, and one of Moriarty's assassins actually saves his life.
Lestrade mentions Inspector Gregson as someone else who has consulted Sherlock. Gregson was another police inspector who often appeared in the stories.
The phone call that John receives claiming that Mrs. Hudson has been shot is a clear reference to the (literal) Swiss messenger from "The Final Problem," who draws Watson away from Holmes with a note about a sick Englishwoman who needs a doctor.
Moriarty's fondness for snipers (they are key components of his plans in this episode as well as "The Great Game") may be a reference to Sebastian Moran, Moriarty's right-hand-man in the stories.
"This phone call, it's - it's my note." In the original story, Holmes left an actual note.
At Sherlock's grave, John calls Sherlock "the best man, and the most human human being, that I've ever known." In the closing words of "The Final Problem," Watson writes that he will always regard Holmes as "the best and wisest man I've ever known."
The 'best and wisest' line got lifted in full for the Sherlock Casebook.
Moriarty's attempt to steal the crown jewels, as well as his being acquitted at trial even though everyone knows he's guilty, are both plot points from the Rathbone/Bruce movie The Adventures of Sherlock Holmes (with Basil Rathbone and Nigel Bruce). Likewise, the climax, in which Moriarty tries to talk Sherlock into jumping off a roof, seems to be based on a similar scene from the Basil Rathbone film The Woman in Green.
Also from The Woman in Green is the sequence where Moriarty enters 221 B Baker street and starts to sneak up the stairs while Sherlock's violin plays in the background. He steps on a squeaky step, the violin stops for a second and then starts again. Moriarty walks up the stairs knowing perfectly well that Sherlock is aware that Moriarty is coming.
All the discussion of Sherlock being "on the side of the angels" is a Shout-Out to the BBC radio plays with John Gielgud and Ralph Richardson (and Orson Welles as Moriarty). In the dramatization of "The Empty House," Watson exclaims "You're the devil, Holmes!" Holmes replies "The only one, however, who has always been on the side of the angels."
Moriarty's plan bears a striking resemblance to The Seven Percent Solution, a non-canon story based around the idea that Holmes invented Moriarty.
A few nods to the canon only appeared in the pilot version of "A Study in Pink":
Since the Mycroft subplot didn't appear in the pilot, an e-mail to "firstname.lastname@example.org" is intended to be this.
The location in which Mike Stamford and Watson dine in the pilot is the real Criterion Bar, in which they first met in the original A Study in Scarlet which led to Watson and Holmes sharing a flat. The 90-minute version showed them having coffee in a park instead, but ensured that the viewer saw the cups bearing the name "Criterion" to retain the reference in spirit.
John's blog entry "The Geek Interpreter" has several references to the original, "The Greek Interpreter". The client's name is Melas, the name of the interpreter in the original and KRATIDES, the fictional organisation, is the name of the Greek prisoner. The names Davenport, Kemp, Latimer, and Sophy are also taken from the original story. Mention of 'a storyline about Latimer, one of the superheroes, defeating two masked terrorists on Shaftesbury Avenue...' might also be a reference to how Sophy originally killed her captors.
"The Speckled Blonde" is an inversion of "The Speckled Band" It looks like she was killed by a snake, but she wasn't.