Continuity mistake: As Chief Wiggum is being interviewed about the fugitive, his arm is shown to be in a cast due to the police van crashing at the Krusty Burger Drive-Thru earlier on. Yet when they get to the hospital, his arm is completely fine. (00:17:30 - 00:19:00)
Factual error: When Chief Wiggum opens the revolver's chamber, there's 5 rounds with one open slot. In the next shot, we see the reverse end of the chamber and it's drawn in such a way to indicate those 5 rounds all have bullets in them still, meaning the empty slot is from when Maggie shot Mr. Burns. But revolvers don't eject spent shells automatically and no one handled the gun after it dropped on the floor. So the empty casing should have still been in there when Wiggum opens it up for the first time.
Factual error: Chief Wiggum visually compares a bullet taken from Homer's gun to the bullet removed from Burns and concludes that they are exactly the same. Excluding the obvious forensic violation, the bullet removed from Burns is still whole and within its casing, meaning that either Maggie took the time to remove the casing and delicately shove it in Mr Burns' bullet wound (which is dumb) or the gun fired the whole casing along with the bullet (which would have blown up the gun).
Plot hole: The director of Radioactive Man says to Bart "we found our new Fall Out Boy. That's what I would say if you weren't an inch too short. NEXT!" But Milhouse is the new Fall Out Boy, and he's blatantly smaller than Bart. Bart was standing right next to Milhouse, so the director could see the difference.
Continuity mistake: After Moe yells at Ralph, there's a woman in the background staring at Moe with an angry look on her face. She turns her head away from him after some seconds, but in the very next shot, she is looking at Moe with the same expression on her face.