The film is set in Gotham City, USA, but the bottle the Joker holds up in his "Smilex" commercial is for a "Moisturising" creme (instead of "moisturizing"), a tip-off that the bottle was created in England, where the film was shot.
On the newspaper that explains the combinations of dangerous products, Odor Eaters is spelled "Odoureaters" (UK spelling in the USA). In the same piece, the first combination is "Deodorant's with Baby Powder" (misplaced apostrophe).
When destroying the paintings in the museum, one of Joker's henchmen puts several hand prints on a painting. Later, we see Joker imitating the statue, and in the background we can see the painting that had supposedly been fingerpainted is not.
During the Joker's Parade, one of the dollar bills that his goons were throwing out to the crowd gets caught on his throne. However, when he tells his goons to put on their masks, it has vanished. It reappears when the Joker pulls out the remote control.
At the beginning of the movie, on the rooftop, Batman pulls out a batarang to catch the fleeing crook. The batarang is matte black, ridged, and hinged in the middle. Once it's around the crook's ankle, though, it's shiny black, smooth and all one piece.
When the Joker is at City Hall with his mime gang, the mime gang opens fire on everyone around. In one shot, two police officers are at the top of the stairs of City Hall, but in the aerial shot, the police officers are gone.
On the rooftop of the Gotham City Cathedral, one of the Joker's henchmen knocks Batman down the stairwell. There is an overhead shot, and even though a bell's yoke is there, no bell is present, supposedly because the Joker made the bell fall all the way down to the bottom a few minutes earlier. A few seconds later, the bell is there, and the henchman and Batman ram each other into it.
After Jack takes the axe and smashes it into the acid vat, we see the acid on his jacket (very visible when we see him running up the stairs to escape the police shooting at him). But when Jack is on the catwalk, with his back to the camera, his coat is perfectly dry.
When Batman piloting the Batwing takes off with the parade balloons. At first there are only four balloons. In a wide shot, five balloons are being carried, including the infant balloon which was seen during the Smylex gas scare. But when Batman sets the balloons free, four balloons are released, not including the infant balloon.
After fighting the Joker's thugs on the street and summoning the Batmobile to escape with Vicki, Batman holds a small remote that controls the car. He raises it to his mouth, gives the command to stop the car and lowers it. In the next shot, his arm is raised to his mouth again.
When the Joker releases the gas during the parade scene, Alexander Knox takes a white dust mask out of the car to prevent him from inhaling the gas. A dust mask only protects you from dust particles and not from gas.
The Cathedral appears to be around 60-70 stories, after Vicki and Joker climb that many stairs, they show no signs of exhaustion nor are they sweaty, the same with Batman. Climbing that many stairs would take more than 10 minutes.
The Bat Signal at the end of the film produces a nice, clean flat image against the sky. A lighted signal such as this, however, would need a backdrop to be cast against to be seen (clouds, for example.)
Even with proper ventilation, it would take at least a few minutes to completely clear a large building filled with gas, but the Joker and his men immediately enter the museum without wearing any gas masks and most of the gas is gone.
The S&W M29 44. magnum with the long barrel that the joker pulls out of his pants would not be powerful enough to take down the bat jet. The size of the barrel would change nothing unless it was a single shot but it isn't.
At the very end, Batman's wire saves himself and Vicki by catching them after they fall. However, the wire is not a bungee cord and has no give, which means that, at the distance and speed that they fell, the shock would've likely killed them both regardless (or at least severely injured them).
The Joker publicly announces that he will be throwing a big parade in the streets of Gotham. The police have been after him throughout the whole movie, yet they're nowhere to be seen during the parade, the easiest time to capture him. (An explanation was written but not included in the film: see trivia.)
When Batman arrives atop the Cathedral at the end, he is attacked by several of the Joker's men. These men appear from nowhere; the Joker indicated on his walkie that he and Vicki were going to be alone and then they climb up alone. The Joker's chopper has not yet arrived, so there is no logical reason why the Joker's men should be atop the Cathedral.
When Batman fights with the Joker's men atop the Cathedral at the end, the fights are very loud and involve all sorts of yelling, punching, kicking, and object breakage. The Joker and Vicki are seen dancing a mere few feet away, yet both are entirely oblivious to Batman's presence until after the fighting is over.
When a thief grabs Martha Wayne's necklace, the pearls all fall separately. But pearl necklaces of any noticeable value invariably have a knot between each pearl to hold them in place and keep from rubbing on and damaging each other. Only the cheapest dime-store version would fall apart that way.
During the fight in the alley after the chase, a goon with two swords appears, and Batman knocks him out. The actor in the Batsuit is clearly not Michael Keaton - the jaw is slightly offset. After this, it is Keaton again.
When Batman goes to blow up Axis Chemicals, he uses pop-up machine guns to cut down a door. We see the guns bouncing up and down with the recoil and with the swaying and bumping of the car, but the bullets still cut a straight clean line without any stray bullets.
(at around 28 mins) In the first meeting of Jack and Batman in Axis chemicals, Batman picks up Jack by the cuffs of his jacket using his forearms against his chest but Batmans gloves have 3 spikes on them and they would have stabbed or hindered his ability to lift him this way.
When Batman drives into the Batcave, Vicki looks behind her to see the cave's door closing. However, the configuration of the windows on the cockpit of the Batmobile would make it impossible for her to actually see that.
After Batman blows up Axis Chemicals he is standing in front of the car as it stops. When the Joker arrives in the helicopter he shines a light at Batman. Then they cut to a shot of the copter leaving with the light off, but when they cut back to Batman it's on again.
When Joker and Vicki are at the stairs to the cathedral it appears to be over 600 feet high, which would mean they would have to climb about 900 steps. There is no possible way they could have climbed all those stairs in the short amount of time issued by the Joker to his helicopter pilot (ten minutes).
Throughout the film, Vicki Vale wears high-heeled shoes. When she visits Bruce Wayne in the Batcave, she's conveniently wearing flat-soled shoes because of the grating Batman's control center is placed upon.
When the Joker grabs hold of the ladder attached to his helicopter when he is trying to escape off the top of the cathedral, it is apparent that he is being held up by a wire as he momentarily takes both hands off the ladder. You can see the wire attached to the front of his belt.
The Joker's revolver should not have been strong enough to bring down the Batwing (it's a holdover from an earlier version of the script, in which the Joker had a tank hiding in one of the parade floats which shot the Batwing down. They weren't able to film it this way for budget reasons though.)
When Vicki examines the platter Bruce used to stop the Joker's bullet, the dimple is on the wrong side. If the platter had been under Bruce's clothes facing that way, its large rim would have created an obvious bulge.
When we first see the Joker as he confronts and kills Grissom, he comes out of the light and a large purple splotch appears on his neck. On the Special Edition DVD of the film, the make-up adviser explains that this was done because some of the white make-up used for the Joker's face got on the coat collar, and rather than taking it off with alcohol, they covered it up with purple shoe polish.