As Grandfather leaves the house after retrieving Gizmo, he is holding the "smokeless ashtray" gift in his hand as he walks out the door. In the outside shot immediately following, the ashtray has disappeared.
In the wide shot after the title "Gremlins" comes up, on the left of the screen you can see a bench with a snowman behind it and no-one is sitting on the bench. In the next shot, two kids are on the bench.
When Mr. Hanson is looking for the changed gremlin after the class leaves, the movie projector is still running and gets to the end of the reel. You can see the film flapping as you would expect, but it's on the feed reel, not on the take up reel on the back as it should be when a film finishes. Not to mention there is no drive on the feed reel, only the take up reel. (The feed reel just turns as film is pulled from it.)
The night after Gizmo multiplies, forming five new Mogwai, he can be seen sleeping in bed next to Billy. In the next shot, a box with the remaining Mogwai next to the bed shows six Mogwai, including Gizmo who can be seen lying next to Stripe. Additionally, a moment later when Billy gets out of bed, Gizmo has vanished from the box.
Near the beginning of the movie, there is a car in the background doing political pitches with one speaker facing forward, and another backwards.
After flashing back to Billy Peltzer, the show the car again, and now only has a single speaker on the front.
In the opening scene at the Chinatown store, Mr. Peltzer squirts toothpaste all over his shirt while demonstrating the Bathroom Buddy. In the next scene, where the kid brings the Mogwai outside of the shop, Mr. Peltzer's shirt is clean and dry.
As Billy's mother attempts to back another gremlin into the microwave at one point, when things cut to showing us the scene from the top of the oven POV, you can see the puppeteer's head ever so briefly before he ducks below the obscuring microwave.
After Stripe's struggle in the Christmas tree, he then smashes though a window. As the camera pulls back from the window, you can see that the bottom of frame has been cut away to allow the puppeteer's hand though the frame.
When Billy's mom takes the picture of Gizmo in the beginning, Gizmo becomes frightened and runs over to Mr. Peltzer. When the scene is slowed, you can clearly see the puppeteer's hand underneath while operating the Gizmo doll.
For a movie supposedly set in late December in a Northern State, all of the Deciduous trees still have leaves on them. The most obvious place to see this is in the opening scene when Billy is trying to start his car in front of his house.
Pete Fountaine uses metal scissors to cut the Christmas lights a Gremlin is hanging off of. Not only are there no sparks from the live wire when this occurs but both sides remain lit despite the circuit's being broken.
In Dorry's Tavern, when Kate turns the ceiling fan on that one Gremlin is hanging on to, in the close up shots, the ceiling fan is turning counter-clockwise, but when Kate jumps over the bar set to attack more of the Gremlins with the camera flash, the ceiling fan is turning clockwise.
Mr. Peltzer tells Billy one of the rules is to keep the gremlin away from water. He tells him, "Don't give him any water to drink and whatever you, do don't give him a bath". We learn later in the film that they multiply with water. Yet they can still walk around in the snow (frozen water) and drink beer and soft drinks (water based) without any signs of multiplication.
Throughout the film, Gizmo clearly says several phrases, such as "bright light" "Mogwai", "Deagle" and the Striped gremlin says "yum yum" when noticing a candy store. Billy hears these phrases throughout the film, yet when Gizmo says "bye Billy", he shouts, "they can talk?" As if he thought the gremlins couldn't talk up until that moment.
In the bar, Kate notices that gremlins are afraid of bright lights when she tries to light the cigarettes of one with a match and the gremlin repeatedly shies back. Yet the whole bar is filled with wafting smoke - supposedly from lit cigarettes and cigars, which we can see gremlins smoking throughout the scene. How were they lit?
In the department store, Stripe attacks Billy with a chainsaw. Billy uses a wooden baseball bat to try and fend him off. The chainsaw would have cut through the bat in no time, but is shown chipping away at the bat for some time.
After the gremlin has tampered with the Sheriff's brakes, the police car nevertheless slows down, displaying its braking lights, as it approaches the next corner. In the next shot, the brakes are not working again. Also, the gremlin is seen ripping off the front hydraulic brake line - any automobile manufactured after 1967 has a dual circuit master cylinder (as mandated by the U.S. Department of Transportation) where one hydraulic brake circuit remains operational when another fails.
When Mrs. Deagle crashes through the window & lands on the snow still in her chair, her legs are completely stiff in the air. This might happen hours later when rigor mortis sets in, but not right away.
When Gizmo is driving the toy car and knocks a bunch of tennis balls and a basketball off a shelf, the camera points straight upward, and we see the balls falling toward the camera and bounce upward. We then cut to Gizmo driving out of the way before any of the balls even hit the floor.
Rand Peltzer and the dog enter the department store at dawn. Just a few minutes later, when Gizmo raises the shade to let the sunlight kill the gremlin, the sunbeam is shining down at a fairly steep angle, almost 45 degrees. This would place the sun much higher in the sky than it would be just after dawn, indeed, higher than it would be even at noon in a Northern state in the middle of winter.