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.
Harsher in Hindsight: In a clip of a news broadcast covering Zilla's rampage through New York, the broadcaster says "...in what city officials are calling the worst act of destruction since the World Trade Tower bombing", referencing the bombing of the WTC in 1993. The film was released only three years before 9/11. Ouch. invoked
Mayor Ebert's put-upon aide Gene walks out on him at the end, and the mayor protests that he can't leave, he needs him. Gene Siskel died of complications from heart surgery a year after this film was released, never returning to Roger Ebert's side on At The Movies.
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.
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 after he's turned away.
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).
And Zilla herself, cuz' you know, she's just a huge raptor.
Reality Ensues: This Godzilla, unlike every other iteration, is not immune to modern weapons.
The Remake: Despite making something new and not respecting the source material Emmerich still has the nerve to call this film a "remake". Fans are, understandably, not amused
Running Gag: All Philippe wanted was a decent cup of coffee.
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.
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)
On the other side of the spectrum, Toho first said "a monster similar to Godzilla ravaged New York at the end of the last century" in GMK, and then put Zilla in Final Wars, where he went down to the Big G himself in ten seconds. Toho did not like what happened with Godzilla in America. Hence the Toho official nickname "Zilla": Toho didn't believe the American Godzilla had any right to have "God" in its name.
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.
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.
What Could Have Been: The original script for the movie had Godzilla being far more like his Japanese self and he was supposed to fight a shape-shifting monster called the Gryphon. Due to the financial success of Independence Day, however, TriStar decided that they wanted their own disaster movie to compete, and so hired Roland Emmerich to direct.