Continuity mistake: When Penguin removes the manhole cover above him right before the Mayor's baby is stolen, he places it right next to the edge of the manhole, but when the baby snatcher jumps in, it is further away.
Audio problem: When Selina returns home from work the first time and Miss Kitty comes in through the window, Selina pours some milk into the cat bowl and Miss Kitty begins to lap the milk (out of shot). Then, without a break in the lapping, Miss Kitty meows again as a cue for one of Selina's lines, but the lapping sounds continue underneath the meow. It's an open-mouth meow, too, and, as far as I know, no cat can both meow and drink at the same time. (00:22:45)
Continuity mistake: The little Bat-Gadget that Batman programs, to knock out the four gang members changes from sailing at head height, to suddenly at about a foot off of the ground between shots, so that the poodle can snatch it. (00:53:00)
Continuity mistake: When Batman is nearing the two clowns juggling flames, the clowns drop their flaming batons twice between shots. (00:13:55)
Revealing mistake: When the clown steals the mayors baby, he does a cartwheel down into the sewers via a manhole. As he falls down the manhole, the "brick" wall of the manhole clearly shakes as the clown jumps down. (00:33:15)
Revealing mistake: When Batman dives off the building with his bat-glider, there are two wires visible coming off his shoulders, most easily visible before he reaches the horizontal. (01:16:45)
Continuity mistake: The position of the fish beside the water cooler changes between shots. (00:51:00)
Continuity mistake: At Wayne Manor, Bruce receives the projected bat signal while in his study. As he stands up, the circle of light hits only the last shelf on the bookshelf behind Bruce. But in the next shot, the light circle is much lower and hits almost the entire bookshelf. (00:13:10)
Visible crew/equipment: A wire can be seen attached to Oswald Cobblepot as his penguins slide him into the water after he dies.
Continuity mistake: When the cats are bringing Selina back to life in the alley, the paw prints they make in the snow around her body are subsequently there / gone. It is not snowing, so fresh snow could not make them vanish. (00:27:35)
Revealing mistake: When saving the Ice Princess, Catwoman throws a chair at Batman. The chair is clearly broken and damaged, even before it makes contact with Batman. (01:14:10)
Continuity mistake: Catwoman is about to land a kick in Batman's face, but from a side angle it hits him square in the chest. (00:57:20)
Continuity mistake: When Catwoman gives Max Schrek the taser-gun kiss, the black wire above them falls down at least four times. (01:50:20)
Continuity mistake: When Selina electrocutes Max while kissing him his head is tilted to the right. Shot changes and it's straight.
Revealing mistake: During the climax, when a rocket hits the "Arctic World" sign and explodes, subsequently causing the polar-bear statue to fall, look closely. The scale of the fire and smoke (particularly the smoke, which looks similar to cigarette-smoke size and not the enormous plumes that such an explosion would create), in addition to the unnatural movements of the alleged large "statue" reveal that this effect was achieved with a rather obvious miniature set.
Audio problem: When the mayor's baby is first stolen, the audience gasps in a very fake way. When the Penguin reveals his plan to steal the first-born sons to the party members later in the movie, they gasp the exact same sound effect. They actually do it a third time, later in Penguin's speech, but it's very quiet.
Other mistake: The security guard clearly throws his gun out of hand even before Catwoman's whip touches it. (00:54:35)
Continuity mistake: The fruit and vegetables (tomatoes, lettuces, eggs etc.) thrown at Oswald at his speech, litter the stage as well. The amount and position of these items laying all over the stage changes between shots. (01:27:10)
Continuity mistake: As Oswald and Max talk beside the water cooler, the shape, size and prominence of the blood on Oswald's shirt, beside his black scarf, changes a few times. (00:49:45)
Continuity mistake: When Catwoman is electrocuting Max, when you see a view of the explosion from above, you can see Max walk off the set through the explosion. It is definitely him, not Catwoman!!! (check the clothing.) (01:50:30)
Answer: The film keeps it ambiguous. There are two ways to look at it. Scenario A- She literally died and is brought back to life by the alley-cats somehow, adapting some of their traits. Or... Scenario B- The trauma of falling and sustaining a head-injury, along with her paranoia after Schreck tried to kill her, has driven her mad, and she uses her connection with cats to build a new persona. (Which is supported by the fact that all the things she claims "kills" her with each of her "nine lives" wouldn't actually have killed her. Ex. Her one fall is broken by the kitty-litter truck. Max doesn't hit her in any vital organs when he shoots her. Etc.) Her destroying her apartment is her lashing out at all the things she used to hold dear- her vision of a normal life, etc. It's symbolic of her purging the past and embracing the future. (Plus, oftentimes when people throw fits, they'll smash stuff up).