Acrophobic Bird: The helicopter pilots seem to always stay at mouth-level of the monster they're attacking, despite the fact that Apaches can fly as high as 20,000 feet above sea level, meaning there was no reason for them to be flying that low to begin with.
Artistic License - Geography: New York doesn't really look like that. Most glaringly, one of the people in the cab says the Brooklyn Bridge is the closest suspension bridge to the Park Avenue Tunnel. The tunnel runs from 33rd Street to 40th Street, and literally every other suspension bridge over the East River is closer.
Bad Vibrations: Whenever Zilla approaches for the first half of the film.
Big Applesauce: If Godzilla usually went after Tokyo, the western equivalent had to attack New York City. (Also marking the second - and not the last - time Roland Emmerich wrecked havoc in the Big Apple.)
Blinded by the Light: The cab's high-beams are used to make Godzilla flinch away from the tunnel mouth.
Breath Weapon: Played with. While the "realistic" Godzilla can't actually exhale radiation, it can roar with such force that it sends parked cars flying from the shock wave. In one case, this causes their gas tanks to explode in a dragonish blast of flame.
And yet they're still more competent than the US military, who arguably cause more damage to Manhattan than Godzilla does!
Chekhov's Gun: Nick notices that Zilla seems to have a bit of an aversion to bright lights after using his camera when Zilla finally shows up for the fish, the flash causing the big guy (or rather big girl, in this movie) to flinch. Nick has Philippe use the taxi's high beams to make Zilla flinch in order to get out of Park Avenue Tunnel in the climax.
Chekhov's Skill: Rather than dig to get to the radiated worms in Chernobyl, Nick used some sort of electrical system that draws them to surface. The same technique is used to draw Zilla out with fish as bait.
Color Wash: The flashbacks that open the film are yellow.
Demoted to Extra: The film is mostly about Nick and Audrey's relationship, the life of a French guy, and Animal Palotti's job, with occasionally a giant lizard thrown in.
Female Monster Surprise: Nick is shocked when a pregnancy test reveals that the giant lizard ransacking NYC is about to have babies. Subverted in that everyone keeps referring to Zilla as a "very unusual he" due to it being asexual.
In Name Only: The monster was nicknamed "GINO" ("Godzilla In Name Only") by Godzilla fans. Toho itself calls him Zilla (see Take That, below).
It Can Think: When Godzilla is tailed by three helicopters, he successfully managed to fool them into thinking he is hiding in a skyscraper, cue him bursting from the building behind them and initiating Curb-Stomp Battle. The final helicopter is destroyed when Godzilla crouches low to prevent being seen by it before devouring said helicopter whole.
When the protagonists are fleeing from Godzilla, he moves ahead of them and takes out the bridge.
Jerkass: Charlie Caiman, who propositions Audrey despite being married, and strings her along with hopes of becoming a reporter while making her do all his work.
Lampshade Hanging: One of the movie's few truly hilarious moments involves a gag pointing out that the beast should be named "Gojira", but Americans got it wrong.
Meaningful Name: A "caiman" is a crocodile-like reptile. See Jerkass above. Roache (pronounced "Rowsh") is a sneaky guy who pops up where he's not wanted and engages in shady activities, befitting one military guy's calling him "Roach".
Merchandise-Driven: All the tie-in merchandise was designed and ready for assembly before a scriptwriter was hired.
Militaries Are Useless: The only time the military accomplishes anything is with the 3 F-18s that both destroy Madison Square Gardens and kill Zilla. Every other time, they fail miserably and either damage other buildings or hit other military units.
Missed Him by That Much: The Jerkass news anchor twice misses seeing Godzilla walk past the station windows because he's too busy complaining on the phone that he doesn't have a good story to cover. One of the army guys overlooks Godzilla in the tunnels because he failed to recognize that the "rocks" in front of him were an eyelid; it opens right behind him and moves after he's turned away.
Monster Is a Mommy: ...about halfway through the film, Nick finds out Zilla had changed gender and laid eggs. If any one of them get out, they'll start a new disaster. It turns out one DID survive, setting up the plot of Godzilla: The Series.
Monumental Damage: Naturally it's often NYC's most-recognizable landmarks that get trashed, either by Godzilla (the MetLife building, the Brooklyn Bridge) or by the forces pursuing him and his offspring (the Chrysler Building, Madison Square Garden).
New York Subway: Zilla burrows into here to make it its new home. Turns out, it did this to get to Madison Square Garden to make it her nesting grounds.
Novelization: By Stephen Molstad, who collaborated with Dean Devlin and Roland Emmerich on their novelizations for Stargate and Independence Day, as well as writing the ID4 prequel novel Silent Zone. An audiobook of the novel, read by Doug Savant, who plays Sgt. O'Neal in the film, was also made.
A Nuclear Error: The opening stock footage of a nuclear test isn't French footage but American test footage out of the Bikini Atoll.
Ominous Crack: The pavement develops a long, spreading crack that runs right under Tatopolous's feet, just before Zilla busts up out of the sewer tunnels in a shower of rubble.
Ramp Jump: The taxi runs over one of Godzilla's toes, which serves as a ramp and sends it flying.
Raptor Attack: Zilla's offspring turns the last half of the film into a fusion between Jurassic Park and Aliens, with Nick and Philippe's team fighting the baby Zillas, which act very closely to Jurassic Park's velociraptors.
Running Gag: All Philippe wanted was a decent cup of coffee.
And on Animal's side, Lucy's gonna kill him.
Scenery Gorn: Taken to some frankly awesome levels - Godzilla jumps through the Met Life building off-screen at one point, leaving a structurally-impossible 'Zilla-shaped hole in the middle.
Scully Box: Alluded to but not used. When Charlie Caiman is about to go live with his co-anchor, a woman who has a good six inches on him, he complains about needing something to match her height. He has to settle for hovering above his chair uncomfortably.
Sensor Suspense: Where the submarine is tracking Godzilla approaching them at a high speed.
Sequel Hook: Didn't get a sequel, but continued in the animated series.
Super-Persistent Predator: The hatchlings pursue humans even when there are still fish lying around waiting to be eaten. Averted with Godzilla itself, as the one behavioral limitation placed on Tristar's version by the trademark-holders was that they couldn't show the Big Guy actually eating anyone: when it does start chasing the cab, it's because it's furious at the destruction of its offspring, not hungry.
Suspiciously Stealthy Predator: Somehow 'Zilla's pursuers manage to lose track of a critter the size of an aircraft carrier on the streets of Manhattan. They soon enough discover that it has been using the City's subway systems to burrow around Manhattan Island.
Roger Ebert: "Now that I've inspired a character in a Godzilla movie, all I really still desire is for several Ingmar Bergman characters to sit in a circle and read my reviews to one another in hushed tones."
Gene Siskel: "If you're going to go to the trouble of putting us in a monster movie, why don't you at least take advantage of having the monster eat or squish us?"
One of the movie's best moments, however, is to its own title; a minor plot point is that the Japanese fishermen who first see the beast call it "Gojira". Footage of this eventually gets out to a certain smarmy reporter, who proceeds to flense it into "Godzilla". Other characters actually call him (or the TV broadcast of him, at least) out on this.
The movie had a quick shot at Jurassic Park, with a major shot of Godzilla's foot going through the ceiling of a museum and stomping a T-Rex statue flat, conveying how much bigger Zilla is than the prehistoric monster. Amusingly, the film was itself (especially their version of the titular monster) was the subject of several TakeThats in later Kaiju films.
Take That, Critics!: In retaliation for giving Stargate and Independence Day negative reviews, director Roland Emmerich had No Celebrities Were Harmed versions of Roger Ebert and Gene Siskel as the Mayor of New York and his aide (respectively). The bizarre thing is, nothing remotely bad happens to either of them (aside from having their city destroyed) and their only real character fault was incompetence (given this is their first monster attack, it's understandable). Both were considerably perplexed by this, with Siskel wondering that if Emmerich was going to go through the trouble of finding lookalikes, why not at least have Zilla squash them? (Also, the extra in the car that is squished early in the film was intended to be a representation of J.D. Lees, editor of G-Fan Magazine.)
Truly Single Parent: In this movie, Godzilla can fertilize itself, like some species of reptiles can.
Truth in Television: Although it wasn't really known at the time, some lizards actually are capable of asexual reproduction via a process called parthenogenesis.
Tsundere: A non-Japanese Type A example with Lucy (ironically enough), who henpecks her husband to no end. Near the end of the monster's defeat, she sees Animal in the news and says in a rather loving way, "I'm going to kill him!"
Voodoo Shark: The reason behind the horizontal, raptor-like design for the monster was in order to make it more "realistic." We'll ignore the improbability of such a lanky, precarious and front-heavy design being better suited for giant size than the heavy, pillar-legged, mountain-shaped original. They decided to continue making it more realistic by making it a mutated iguana instead of a dinosaur, thereby completely negating the entire point behind the raptor-shape in the first place. And the Square/Cube Law is being completely ignored either way.