Plot hole: Batman finds at Belson's home a handwritten list of substances needed for a transplant. Why would Belson, a medical authority in the field that knows those things thoroughly, even write a list down? If it's for Freeze to procure him the material, why would it be at his house and not with him, and when did he even have time to be at home to do that, since he has been kidnapped by Freeze when he was driving on the outskirts of Gotham and taken to the oil rig? If he had time to go home, he at least would have changed clothes, he's still in the tuxedo he wore when he was kidnapped the night before.
Continuity mistake: The heels on Barbara's shoes appear and disappear throughout the movie.
Continuity mistake: While Batman and Robin are at Arbagast's home, the phone rings so Robin gives Arbagast the handset. When they realise it's Belson on the line, in a closeup Robin clicks the machine base to record the conversation. The problem is that in this closeup the handset is still in its base, even though Arbagast is still holding it talking to Belson. (00:42:45)
Continuity mistake: On the burning oil rig, when Robin and Barbara pull Mr. Freeze out from underneath the rubble, Barbara is missing her shoe and blue sock in this shot. (00:54:20)
Other mistake: While the fire on the oil rig burns all around, Mr. Freeze grabs Belson, then orders him to go find Barbara (to do the transplant). When it cuts to a closeup of Freeze, the reflections of Belson on Freeze's eyeglass lenses are not reversed as they should be. (00:53:00)
Plot hole: There are 17 possible donors for Mrs. Fries. Of all of them, instantly and with no explanation, the doctor chooses Barbara Gordon. She was not the first alphabetically, from the phone call they find out she has a roommate, a boyfriend, she is out for dinner at the moment and is going out of town and of course, REALLY huge warning sign, she is the daughter of the police commissioner (hard to ignore that in Gotham, and Freeze certainly knows him first-hand!). There's no reason why to kidnap her, plenty of reasons not to, and zero reasons given why they did not try any of the other 16 names on the list.
Suggested correction: They did explain exactly why they picked Barbara. The names on the list weren't all perfect matches. They were looking for someone "who's approximately the same weight, height, and female." When he sees Barbara's file, she's a perfect match. The fact that it's Jim Gordon's daughter is of no concern to them, assuming they know it's his daughter since Gotham is said to have a population of 10 million.
You are quoting the movie but then describing something radically different from what is shown (which I understand, since what is shown makes little sense). The list of 18 people (one being Mr. Freeze's wife) is the one that appears exactly *after* he has refined it for "approximately the same weight, height, and female." Everyone on that list should be a perfect match, otherwise there was no point at all in showing him writing F, 5'2", 116 lbs, AB Negative in the search fields and get search results for it. If it's not, he has no way to tell at first glance.
Continuity mistake: Batman is hit by a falling piece of debris as he tries to save Freeze. He seemingly lets just go a button (or something) of the bat-rope gun, since it's still in hand, but a moment later he is dramatically reaching down calling Freeze's name and the hand is empty. He then stands, bat-rope in hand. (00:59:05)
Other mistake: Freeze uses his gun to extinguish the fires on top of the elevator, and that somehow gets it back to work, but it does not really make sense. His gun greatly weakens metal, as shown earlier, to the point of breaking it, and he did not shoot the engine of the elevator, which is what was shown malfunctioning.
Continuity mistake: In the room where Nora is stored, Barbara smashes with a pipe the bottle on the table. The liquid is spread to one side in the first shot, in a circle in the following. In the next shots, more inconsistencies appear (liquid appears to be leaking off the table, only to show later a half-circle nowhere near to any of the table edges). (00:34:45)
Continuity mistake: Barbara escapes traveling inside what is shown as a straight ventilation pipe, the closest intersection to her left being practically all the way to the back of the room with one to the right somewhere in between. Magically once Belson slips past the grate she has a left opening from a T juncture to escape into, right behind her. (00:33:25)
Continuity mistake: Barbara slips through the air conditioning conduit into the room next to her cell. Freeze uses his gun to cut her escape path. We see that the icy spikes are past the vent, since Belson opens a grid right before them, but when Barbara kicks the stalagmites loose, Belson pops up behind (or from) them. (00:33:30)
Other mistake: Freeze gets in sight of the toll booth (we see the headlights from his truck on the asphalt before it), and there's a car in the outer lane he is travelling in, but he has a (too) long and uninterrupted run in that lane. That aside, the road goes from having 4 lanes to 3 lanes and back to 4 lanes, and everyone is going at mad speed having barriers this close (especially with the toll booth not being automated). (00:27:00)
Factual error: The plot revolves on finding a matching organ compatible with Nora. Practically the only issue presented about it is the blood type compatibility, so let's go by that logic. In a misconception happening in so many works of fiction, the fact that she is an AB - is listed as a terrible problem because of how rare it is. It is indeed the rarest blood type in most parts of the world, but it's also the universal receiver and compatible with every other group.
Other mistake: When Dr. Belson searches the blood donor database, a lot of weird stuff happens; first of all, he is searching it by name, and it's the name of Mr. Freeze's wife. Since for some weird reason she is shown later being in the donors database (doubtful that someone with an incurable disease would be a donor!), it should return the information about her, but it does not. Secondly, he is opening the database search again once Freeze promises the gold vein, and the form shows already the personal data of Nora but without her name this time, and including the blood type, which he types only in the next close-up. (00:20:50)
Continuity mistake: During the fundraiser, glasses of relevant guests (the woman in the red dress when she is introduced to Bruce, the stock broker when he is dragged away) disappear conveniently off-camera. It's quite a stretch to assume they found a waiter or a place to place the glass when they were not in frame (especially in the group of three women in front of the doors).
Continuity mistake: Batgirl roundhouse kicks one guy in the park. In the background, the victim has her left arm down, but it's up in the following shot. In that shot her walkman is also missing the cable linking the headset to it (it is there when Batgirl helps her stand later, same camera distance). (00:12:35)
Continuity mistake: The movie gets a lot of details right when it shows the establishing shot of the exterior of the fundraiser event Bruce Wayne goes to after the initial action; perfect continuity with the car models and people in the crowd ascending the stairs. But somehow, it has three people standing in front of the teal car, instead of 'just' two. (00:11:40)
Continuity mistake: Belson releases the mooring line to take off with the boat, and we see the rope slip away in close-up. Too bad that in the shot that follows the ignition, the boat is still moored, or at least the line is still in place. (00:55:35)
Other mistake: Freeze's hideout is in an abandoned oil rig, with its tanks full to the brim of oil. You'd think the company owning it would have drained the millions of dollars' worth of content first, before 'abandoning' it.
Continuity mistake: After beating the two robbers up, Batman and Robin dump them in a pile of trash on the sidewalk. That pile of trash was not there when the car turned the corner and crashed in front of the grocery store. (00:10:35)
