Creator Thumbprint: As listed at the trope entry, there's unusual hands, German Expressionism, dogs, Goth Spirals, and stripes. To a lesser extent, snowy settings, especially if Christmas is involved. Above all these, he loves protagonists who are outcasts in some way.
He also seems to like casting his disproportionately attractive girlfriends alongside much hotter male leads. He also seems to like killing off (Mars Attacks! (shot), Sleepy Hollow (iron maiden), Sweeney Todd (roasted alive)) or at least giving a less-than happy ending to said girlfriends (Big Fish (spinster cat lady), Alice in Wonderland (exiled) and Planet Of The Apes (then-girlfriend Lisa Marie and future-wife Helena Bonham-Carter were implied to be erased from history).
Darker and Edgier: Pee-wee's Big Adventure was actually quite colorful and kid-friendly, at least compared to Burton's later work. Beetlejuice began to introduce darker and more adult elements, and Burton's output only increased in edginess from that point until we got the downright nihilistic Batman Returns in 1992. Following the ugly backlash to that last film, Burton retreated from his misanthropy a bit and returned to his comedic roots with Ed Wood and Mars Attacks!
Dark Is Not Evil: It can be benign, world-weary, friendly, helpful — even lovable. That said, when dark is evil, or at least angry, watch out.
Deliberately Monochrome: Ed Wood is in outright black and white, Halloween Town in The Nightmare Before Christmas only has touches of color, Sweeney Todd is shot in mostly desaturated colors, and the living world in Corpse Bride is much grayer than the world of the dead.
Dutch Angle: Crops up in a majority of his films, ranging from blatantly obvious to the blink-and-you'll-miss-it variety. For particularly noteworthy examples, please examine Batman and The Nightmare Before Christmas''.
German Expressionism: His aesthetic style is completely derived from this film movement.
Goth Spirals: One of his signature motifs. Look closely at the page picture; he even has one in his hair.
The Grotesque: Edward Scissorhands is an excellent modern example. The Penguin in Batman Returns is a cruel subversion — he masquerades as this to gain sympathy and make a grab for power.
Loners Are Freaks: It seems like he dedicated almost all of his movies to invert the trope.
Our Monsters Are Different: Many of Burton's protagonists consist of characters who are viewed as terrifying to other characters, but they're actually the heroes of the piece, even if what they do is not in tune with everyone else.
Person As Verb: "Burtonesque" is often used to describe something with a gothic, quirky style.
Playing Against Type: While Burton's choice to cast Michael Keaton as Batman was surprising at the time (see trope entry), it's his work with Johnny Depp that's most famous for invoking this trope. Depp was eager to move on from 21 Jump Street, which made him famous but threatened to typecast him, and doing Edward Scissorhands (and, for John Waters, Cry-Baby) in 1990 was incredibly effective. "Eccentric protagonist" has become Depp's "type" for Burton.
Recycled Trailer Music: The scores for Pee-Wee's Big Adventure, Edward Scissorhands and The Nightmare Before Christmas turn up as the music to trailers for other films quite often.
Scenery Porn: It's not for nothing Batman, Sleepy Hollow, and Sweeney Todd won art direction Oscars.