Acting for Two: Natalie Dormer effectively has to play three different characters: Irene Adler in flashbacks, Irene Adler in the present with PTSD, and Moriarty. Of course, the former two don't actually exist.
Joan's brother in "The Leviathan" is called Oren, named after Lucy Liu's character O-Ren Ishii of Kill Bill.
Moran is a big Arsenal fan; Vinnie Jones used to be a professional footballer. Holmes' disdain for this could also be a nod to the fact that Jonny Lee Miller supports Chelsea, Arsenal's long-time rival.
In "Snow Angels", Sherlock sarcastically says that his companion "holds several black belts" to a security guard wielding a knife, a reference to Lucy Liu's talents in martial arts.
In "Dead Man's Switch" Sherlock says to Gregson "You have daughters, do you not." The writers' Twitter feed state that it was a nod to Aidan Quinn, who has two daughters.
Again in "Dead Man's Switch" we have a blackmailer called Henry 8, referencing Henry VIII of England. Natalie Dormer, who plays Irene Adler, had her first major role in television as Anne Boleyn, the second Henry VIII spouse in The Tudors.
In "All In The Family" we learn that Joan Watson was born and raised in Queens, NYC. Lucy Liu too.
"The Many Mouths of Aaron Colville" is not the first time we've seen Jonny Lee Miller in a prom dress.
Sherlock, just like his actor, is a major fan of football.
Fandom Rivalry is more accurate. The shows themselves are hardly competing financially, but fans of each respective show continue to argue over which is better, despite the fact that the shows, while sharing a similar premise, are vastly different; Elementary is more of a procedural, while Sherlock is more of a mystery drama. This hasn't been helped much by Steven Moffat and other Sherlock alumni sharing their less than shining opinions of Elementary.
It doesn't help that CBS asked BBC for the right to create a US remake of Sherlock and were turned down (Sherlock was doing very well on BBC America). They then announced that they would be making Elementary. It's understandable that some felt that CBS were making an American Expy with just enough differences that BBC couldn't call foul.
Fake American: Sherlock occasionally does this while trying to obtain certain details, like the locations of persons of interest. Jonny Lee Miller's accent is surprisingly flawless... but then, anyone who saw Eli Stone or watched the fifth season of Dexter would already know this.