Revealing mistake: When the Riddler projects the question mark above the bat signal, two beams can be seen producing the image, except they are slowly tracing the perimeter. If these beams are producing the image, it should only be visible where the beams are touching, and not everywhere else. In order to produce an image and not just two points, beams would need to travel extremely fast to trace the perimeter without leaving gaps. (much the same way images in a lasershow are produced). (01:33:10)
Continuity mistake: When Bruce is reading the riddle about vowels, it starts out "We're five little items of an everyday sort." He pushes the button to read the second part of the riddle. Then when Bruce walks around, the first part of the riddle can be seen again. (01:33:25)
Continuity mistake: When Bruce wakes up after getting shot in the head by Two-Face, there is a "BW" (Bruce Wayne) on his pillow next to his left ear. But after Alfred asks, "How are you, young man?", the "BW" is gone. (01:35:55)
Continuity mistake: After the Bat-Boat is destroyed, it changes to a normal wooden boat. (01:41:50)
Revealing mistake: When Dr. Meridian first drops out of the tube, her skirt flies up. However, when she is falling in the shaft, her skirt is down and barely moving. (01:46:55)
Continuity mistake: At the end of the film right after Batman has saved Chase and Robin from falling and Two-Face appears on the ledge, you can see that in one shot Chase has lost her shoes and in the next shot she's wearing black sneakers. (01:48:00)
Continuity mistake: Right before The Riddler is about to kill Robin and Dr. Chase, his finger changes position on the button on his cane between shots. In one shot it is next to the button but in the other it is suddenly completely covering the button. (01:57:15)
Continuity mistake: In the scene in which Edward is first testing his "box", watch Ed's left shoulder. He goes to turn off the machine, then says his line "What a rush." The shot cuts to Ed's boss briefly. When the shot cuts back to Ed, his left shoulder strap, which was present in every other shot, is missing. He even reaches to take the strap off as he starts to say "A very surprising side effect." Apparently, he didn't even notice that it had fallen off.
Continuity mistake: When Riddler and Two Face are stealing diamonds, the Riddler is holding a large diamond. Two Face comes over to him and produces an even bigger one. However, if you look closely at the shots, the diamond the Riddler holds gets smaller in one cut, and Two Face then pulls out the diamond Riddler was just holding...
Continuity mistake: When Two-Face and his goons are fleeing Nygmatech after their break-in, as they're running down the stairs there are 3 men in identical blue suits running just ahead of Two-Face. In the next shot they're nowhere to be found.
Continuity mistake: When Batman jumps up into the Riddler's lair, he's gone up the outside of the shaft - we see a shot from his point of view right up to the point of entry, and he's right at the edge. But when he jumps back into the shaft, despite not having moved, he's suddenly right in the middle of it.
Continuity mistake: When Edward Nygma meets Bruce Wayne his hair is messed up around his forehead. A shot later it's nicely combed.
Continuity mistake: When Bruce asks the doorman where Dr. Meridian is a man walks down the steps and leaves. Shot changes and he is walking down the steps again.
Other mistake: The shot showing the vault swinging back to his location is the same that the one where the vault is swinging out, but going backwards.
Continuity mistake: In the fight scene outside the elevator at the beginning of the movie, there are at least two shots where you see Batman's back without his cape in the way. There are no items whatsoever attached to the back of his belt. Subsequently he manages to pull the stun device and the grappling gun with grappling hook from behind his costume. A previous corrector proposed that the devices could not be seen because they are concealed in "pouches" on Batman's belt. Close inspection of the scene shows that there are no pouches of any kind on his back or concealed in his cape.
Continuity mistake: Two-Face and the Riddler are in Two-Face's lair; the Riddler has just said, "Then I taught my doggie a new trick; how to map the human mind." In the establishing shot, the tube is in left hand; after the cut, it's in his right hand.
Revealing mistake: When Batman is hanging from the chain that is connected to Two-Face's helicopter, you can see a dark area at the bottom of the screen. The dark area is a shadow, and you can tell it's green screen. And yes I know he's hanging off a helicopter.
Continuity mistake: When Dick Grayson is showing off his fancy laundry skills, he splashes water onto the washing machine and dryer. In the last shot when he mops up the mess, the water is gone from one side of the machines.
Continuity mistake: When Edward Nygma is in his apartment after killing Stickley, we see him pasting letters for the next riddle involving the match head. However, when Bruce and Chase are in her office looking at the riddle after it was delivered, you can see the letters are in different positions than the earlier shot.
Continuity mistake: Bruce's hair is much flatter and messier in the garage than in the previous scene outside when Dick is leaving, even though the garage scene happens directly after.
Answer: It's called "Smash it up" by The Offspring.