Creator Thumbprint: Only "The Empty Child", "The Doctor Dances" and "The Beast Below" don't involve the Timey-Wimey Ball in some way, and of those, only "The Beast Below" doesn't have time travel integral to the plot (as opposed to just landing the TARDIS there).
He seems to like using Clarke's Third Law as a plot device: apparently supernatural and bizarre (and terrifying) events are eventually explained as the result of malfunctioning advanced technology in a more primitive setting. This is used in "The Empty Child / The Doctor Dances", "The Girl in the Fireplace" and "Day of the Moon".
Official Fan-Submitted Content: Aside from the typical contests from the likes of Blue Peter that surround Doctor Who, there has been some Unofficial fan content. Fans will occasionally ask him a question at a convention or on Twitter, and he'll express some genuine interest in trying to put the answer into the show. He's said he tried to work in an explanation to a question about Weeping Angels and mirrors, but it was cut. Similarly, a fan asked if a pregnant Time Lady's unborn baby would regenerate if she has to regenerate, and it really seemed to pique his interest.
One of Us: Freely admits he first petitioned for the job of Doctor Who showrunner when he was 7 and jokes that his entire writing career has merely been a clever ploy to get there.
Signature Style: Most of his episodes are more psychologically scary, leaving you cowering behind the couch despite a body count of zero. He tends to invoke the Uncanny Valley quite often. His monsters involve masks (Clockwork Men, Empty Child) and statues/mannequins (Weeping Angels, Smilers).
Also has a tendency to make the most innocuous things absolutely terrifying, like the voice of a child asking for his mother, clock ticks, statues, shadows, birdsong, mirrors, a crack in the wall... and now, after making almost everything else scary, he goes Up to Eleven and makes silence itself terrifying. He wants to leave his audience with no place to hide.
He's also mentioned Florida several times in a really short time frame. In "The Big Bang", the Doctor and Amy visit "Space Florida." "The Impossible Astronaut" and "Day of the Moon" are partially set in and around Cape Kennedy.
Scary spacesuits have also shown up in at least two different stories.
Moffat's stories tend to be about time travel in some way rather than just using time travel as a plot device.
As mentioned above, he doesn't like Technobabble, and tends to try and subvert it when possible.
In "A Christmas Carol" when he's trying to explain why the flying fish like Abigail's singing, he begins to talk about stimulating ice crystals, during which a fish bites him. ("Look, the fish like singing, now shut up!")
Also, in "The Doctor's Wife" (which was written by Neil Gaiman but had finishing touches by Moffat), when the Doctor tries to explain why he can't put the TARDIS matrix in another human body, the TARDIS itself begins to spark almost spitefully (Rory's fault, but it was timed way too well), so the Doctor gives up and says "All right, yes, it's Spacey-Wacey."
Since his current job involves Time Travel, and all the confusion that naturally occurs, he's developed a tendency to have characters change their names, often giving them a "Young Name" and an "Old Name." This happened with Amelia Pond/Amy Pond, and happened again with Melody Pond/Mels/River Song. In fact, it harkens back to "The Girl in the Fireplace," with Reinette/Madame du Pompadour.
While a lot of these aren't specifically written by Moffat, as producer he's had some say in them (approval, etc.) That being said, there have been a lot of Amy Pond doubles floating around. Let's count: Time-Shifted Amy/Amelia ("The Big Bang"), Time-Slipped Amy ("Space," "Time," "The Girl Who Waited"), Teselecta Amy ("Let's Kill Hitler"). Let's hope Karen Gillan gets time and a half for all the Acting for Two she did.
Furthermore, this tendency also shows in his decision to turn the fifty-year-long question of the Doctor's real name into a major plot point in Series 6, that there's a Dark Secret behind why the Doctor purposely conceals his true identity and that he chose to adopt the name of "The Doctor" because he considers it to be a promise.
Realising that River Song is a polarising to some fans, he and Neil Gaiman gleefully announced an episode titled "The Doctor's Wife". The section of the fandom that didn't like River went berserk ... and River did not appear in the episode.
He also loves to straight-up lie about his plans for the show. Reached epic heights with the 50th anniversary, which he claimed would feature no classic Doctors, did not involve Tom Baker in any way, wouldn't provide any peeks at Peter Capaldi's upcoming incarnation of the Doctor, and wouldn't really involve much if any of the show's canon from before 2005. The climax of the episode was a scene of every single Doctor, even Capaldi, working together to save the Time Lords and Gallifrey, thereby restoring one of the biggest parts of Classic Who's canon that the new series changed, followed by a quite heartwarming cameo by Tom Baker. Well played.
He's also fandom-savvy enough (he's been at it long enough, after all) to know exactly the right things to say and do to whip his substantial Hate Dumb up into a lather, which he does with some regularity. Predicting what he'll be bashed for this week makes for an entertaining spectator sport.
Word of Gay: Has stated that River is bi, and that the Doctor has no real concept of human sexual preference for one gender over another. Both things were hinted at in his stories, but only became explicit on his twitter.
Phones seem to be one of his things, they tend to be featured prominently and as integral plot devices, both Sherlock "A Study In Pink" and Doctor Who "The Eleventh Hour" feature mobile smart phones as plot devices.
"The Impossible Astronaut" also has phones as plot devices, both mobile and land-line.
The Empty Child from the episode of the same name is fond of placing unearthly calls, even to the TARDIS' fake phone.
"The Beast Below"'s cliffhanger involves a phone call, and "The Pandorica Opens" is kicked into gear by a phone call as well.
Coupling has several episodes based around misunderstandings with phones.
Deconstructor Fleet: This was hinted at with "The Curse of Fatal Death." Both Matt Smith's Doctor Who and Sherlock are defiantly 'anti-myth', as it were. Sherlock is a self-aggrandizing genius who gets a taste of his medicine in "The Reichenbach Fall", when every character he has insulted throughout the series conspire to ruin his reputation and drive him to apparent suicide. The Doctor is a bit friendlier, but makes no bones about being a "mass-murdering psychopath" who often dooms people by offering them an escape from real life. The overall message seems to be that there are no clean "heroes" and that myths are dangerous, though Moffat concedes the childlike wonderment of myths.
Girl on Girl Is Hot: Any male character (except the Doctor) will become near-comatose at even the mention of lesbianism. Oddly enough, this includes his version of John Watson, whose sister is a lesbian.
Last Name Basis: He's noticed that fans frequently refer to him as merely "Moffat," with some confusion (he blames Matt for starting the whole thing.) He then snarked, since Jenna Coleman dropped the "Louise" out of her name after the end of series 7, he was going to take a page from her, and would now call himself, "Steven Fat."