When The Penguin is in the Hall of Records, we see him through a window and it's snowing outside. In the next shot we see the Batmobile come around the corner and it's not snowing, but in a shot of Batman from inside the Batmobile, it's snowing again.
When Catwoman breaks into Schrek's Department Store she jumps off of the trampoline, and when she lands, her whip touches the floor. But in the next shot she still has the whip wrapped around her, and then starts to unwrap it.
During the 'Lawn Dart' scene Penguin's left gray top coat sleeve is sagging off his shoulder revealing the lapel of his black tuxedo coat. As he stands and waves his hands about, his top coat covers the lapel. When the Penguin cackles at Batman's frame-up his lapel is revealed again.
In the original staff meeting just before the first tree lighting ceremony, when Selina Kyle is serving coffee, the camera angle changes as she walks around Max Shreck and she ends up walking around him twice.
When Selina comes home from the office and listens to her answering machine messages saying she has to go back to the office to get the Bruce Wayne file, she is wearing a brown dress suit. But when she comes home after being brought back to life by the cats, she is wearing a gray dress.
After "rescuing" the mayor's baby, the Penguin rides one of his duck boats up on a scissor lift. The boat is shown to have short walls on the side and the duck's head is clearly the highest point on the boat. But when Penguin rises out of the manhole, his feet are nearly level with the ground, implying either he had another lift inside the boat or the rest of the boat was crushed against the sewer ceiling.
When the Penguin attempts to kill Catwoman using his flying umbrella, when he delivers the line "good bye my unintended, go to heaven" his hands are empty. However in the next shot he has a glass in his hand.
Just prior to the explosion of Shreck's, Catwoman famously back flips her way toward Batman and Penguin, who are directly in front of either side of the camera. On the VHS version, you can clearly see Michelle Pfeiffer, in catsuit, hunched down at the bottom of the screen, ready to jump up and "meow" for the camera once her acrobatic stunt double is out of the shot. Although the film's final editing didn't require the scene to be shot this way, Ms. Pfeiffer and her double still ended up in the same frame.
At the end of the movie when Penguin is accompanied into the water by penguins, when his body floats into the water, you can see the skid he's on slide out from under him and out of frame on the left middle of the screen.
If the guided Batarang had enough inertia to knock out the previous targets, the poodle that jumped up to intercept it before hitting the "Marie Antoinette" gang member would have just become part of the projectile.
When Bruce plays the CD, just for fun he "scratches" it back and forth like a record a few times, and it sounds just like record scratches. Although the technology to do so in real life was not invented until much later, Batman is shown having several pieces of advanced technology that was impossible for the time (for example, the self propelled Batarang and the Batmobile's armor). There is no reason to believe Wayne did not invent it himself.
Catwoman engages Batman in a rooftop fight minutes after she destroys Shreck's Department store. But Shreck's Department store - in perfect condition - can be seen behind Batman while he is hanging from Catwoman's whip after falling over a ledge. However, Max Shreck is a very prominent, wealthy businessman in Gotham, so it is possible he has multiple department stores in the city.
When Batman is on the roof and his cape becomes a glider, you can see that the glider is separate from his cape as his cape falls off behind him. When he floats down the glider cape once again resembles his regular cape.
After Penguin slides out of the duck vehicle from his failed mayor campaign he's wearing his black tuxedo coat under his gray top coat. After he pets the penguins in the next scene he's only wearing his gray coat over his vest and shirt.
When the baby is stolen during the press conference and the thief jumps down the man hole into the sewer you see the fake stone just beneath the surface of the man hole shake. This would suggest that it is made of plastic or rubber.
The same switch is used in the Batmobile to turn on the rocket booster in the beginning and activate the Batmissile during the later chase. The Batmissile switch finally works, but it is in the off position.
In the electrocution/explosion finale (on a wide long shot), a stuntman/technician can be seen running and hunched over from where the main character(s) were, to the viewers right, behind another part of the generator.