BBC Sherlock:
Sherlock Holmes: INTP
John Watson: ISFJ
Mycroft Holmes: INTJ
Gregory Lestrade: ISTP
Molly Hooper: ISFP
James Moriarty: ENTP
Irene Adler: ENTJ
Charles Magnussen: INTJ
Mrs. Hudson: ESFJ (?)
James Sholto: ISTJ
Mary Watson: ESTP
Donovan: ESTJ (?)
Anderson: ISTJ (?)
Mrs. Holmes: ENFP (?) - not enough info
Mr. Holmes: ISTP (?) - not enough info
Hm, clearly I was subconsciously trying to avoid type overlap, which creates a too perfect analysis... Although I adore the idea that Mycroft and Magnussen are the same type (didn't notice this until I tried to type Magnussen)--if that is truly the case, Sherlock might rail against him like he does because he is reminded of his brother. In their interaction, he is still trying to prove himself as an equal and still comes up short. Heartbreaking. I would love further debate, especially about minor characters that I brushed over. I am pretty certain about the main characters, especially Sherlock and Mycroft and can offer thorough analysis if anyone is interested, but there is already a lot written on these types online and on this forum and I'm not sure I can offer a significant amount of new insight. I personally firmly believe that Sherlock is not an INTJ. There are a plethora of reasons; what follows is simply my favorite:
-Fe vs. Fi
Because it is pretty much universally accepted that Sherlock's Feeling function is either a tertiary or an inferior function, this function is often completely overlooked when typing. However, as an INTP who struggles with my feeling function, I can attest to the fact that it is the inferior Fe function that challenges and often betrays the INTP. Here are characteristics about Fi and Fe that support my conclusion that Sherlock has Inferior Fe (and not Fi):
Fe: want to improve the world, have trouble hiding their emotions, highly value close friendships, struggle to understand their own emotions, want to make others happy, are forgiving, can be too controlling of others, are vengeful against wrong-doers, admit when they’re wrong, are good listeners, are expressive, are attentive to the reactions of other people, ultimately desire to be loved
Fi: are concerned with their internal values, believe in good vs. bad and right vs. wrong, are emotionally intelligent (usually know how they feel about things), constantly evaluate whether their choices match up to their values, are empathetic, are accepting of others, are merciful, prefer to keep their emotions private, want to stay true to themselves, take things personally, ultimately desire self-actualization
A common argument as to why Sherlock has Fi (and is therefore an INTJ) might go like this: Sherlock calls himself a sociopath and makes it very clear that he is not solving cases in order to help people, but rather for the fun of solving puzzles. He seems unconcerned with John’s feelings and claims to have no desire for “harmony.†Very Fi, right?
He’s lying. Probably in an attempt to be more like his (ostensibly) more stable, more intelligent brother who does have Fi in his function stack. I've classified Mycroft as an INTJ.
Therefore I posit that Sherlock is actually an INTP with a strong Ti and an underdeveloped Fe trying to rationalize his often overwhelming feelings with logic. He has frequent tantrums and breakdowns (you can always read his emotions right on his face), solving the puzzle isn’t always enough for him when innocents get hurt (when the old woman dies during his case S1E3; when he inadvertently hurts Molly at the Christmas party S2E1), and he is more than willing to sacrifice his own life and health and general well-being for the life of another (John).
He has no personal values (i.e. lying is always wrong, life is always sacred, never hurt your family). A Fi user would. And while some might argue that he demonstrates his true values in the way he approaches his cases, I’d like to argue that an obsession with uncovering the truth is not a value; he uncovers the truth because he finds it fascinating and it keeps him from going insane in his own head. A good example of his lack of “values†is the very manner with which he puts himself in danger for John: he believes that
John’s life is sacred, not that life is sacred in general. Period. After all, he ruthlessly kills Magnussen and seems to feel no guilt.
To briefly elaborate on how conflicts for people struggling with Fi/Fe might be dramatized/fictionalized:
As I've stated, Fi is associated with self-knowledge. A Fi related journey would be one by which a character(s) learns or fails to learn about the self. (i.e. Siddhartha, Harry Potter, Alice in Wonderland, The Lion King, Hamlet, Macbeth, Richard III, The Odyssey, Crime and Punishment, The Catcher in the Rye, 1984, Invisible Man, The Stranger, The Picture of Dorian Gray, Oedipus, Frankenstein, The Color Purple, A Clockwork Orange, Mad Men, Breaking Bad, Jane the Virgin.)
Fe is associated with interpersonal relationships. A Fe related journey would be one by which a character(s) learns something fundamental about humanity or fails to do so, usually through struggling with and then reconciling with or coming to terms with another (or other) characters. (i.e. Titanic, Black Panther, Frozen, Beauty and the Beast, Shrek, Up, The Great Gatsby, Othello, Romeo and Juliet, Julius Caesar, King Lear, Twelfth Night, The Taming of the Shrew, One Hundred Years of Solitude, Lolita, Wuthering Heights, Pride and Prejudice, Jane Eyre, Beloved, Lord of the Flies, The Scarlet Letter, On the Road, Gone With the Wind, The Walking Dead, Supernatural, Criminal Minds.)
Overlap can occur. Occasionally, settling Fe problems leads to Fi growth (Call Me By Your Name) or settling Fi problems leads to Fe growth (Lady Bird) and some stories include both types of problems, especially ensemble pieces when different characters are struggling with different things (i.e. Orange is the New Black).
IMHO, BBC's Sherlock is transparently concerned with testing Sherlock's Fe and actually falls short in its exploration of what it might look like for Sherlock and John to experience individual growth. Within the show, Sherlock has to choose between his life and John's life (s2/s4), his brother's life and John's life (s4), and John's happiness and his own happiness (s3). He always chooses John, even if it means halting his own growth (like when he rekindles his drug dependency in s4). In John, he is confronted with a man who ALSO has trouble trusting, forgiving, and expressing his love for other people (ahhhhh, double Fe conflict

). John has interpersonal problems as well, which Sherlock fixes instantly by just being himself. No personal growth required.
As far as I can discern, neither character has any interest in self-actualization whatsoever or in “being true to himself.†Sherlock's interest is primarily in figuring out the external world in his own head (Ti/Ne) just for the hell of it. John's interest is in "fixing" himself. Which is why at the beginning of the series, they are both promptly floored by the realization that they might be capable of having a companion to love and be loved by in return. That is the focus of the show; they become better because they have each other, and fall apart when they don't. The conclusion of Season 4 is a declaration that they belong together, and that all can be well in the world when they are.
P.S. Based on my typings of the characters, this are all the characters with Fe in their function stacks: Sherlock Holmes (INTP), John Watson (ISFJ), Gregory Lestrade (ISTP), James Moriarty (ENTP), Mrs. Hudson (ESFJ), Mary Watson (ESTP). Notice the importance of these characters in the show.