Continuity mistake: When the video of Kimmy plays on the school announcements, and she talks about Duncan being hung up on Veronica, there is a cut to Duncan reacting with raised eyebrows in the foreground, while Veronica can be seen in the background with a passive expression. Then it cuts to a close-up of Veronica having a visible reaction with her whole face falling. But since Veronica and Duncan were reacting to the same thing, Veronica's reaction should have been visible in the first shot as well.
Continuity mistake: Deputy Leo is at the police station when a woman reports her daughter is missing, but moments later when Keith tracks Veronica to a storage/warehouse, Leo is playing with his band and doesn't seem to know what is going on and Keith has to tell him that a girl is kidnapped.
Other mistake: When Beaver takes the iPod/camera, the pictures are left on Veronica's laptop. This would only be possible if the pictures were copied onto the computer. It would have taken a considerable amount of time for all of the two thousand photos to be copied. There is the possibility that Beaver copied the three or so photos of Logan, but considering his anger, that's hardly something that he would take the time to do.
Plot hole: When Veronica tells Mac that she was switched at birth with a girl who was born the day after her in the same hospital, Mac repeats that the girl's birthday is the day after hers, as if she already knows what she's about to read on the paper that Veronica gives her, which is that she was switched at birth with Madison. That would make sense, because Madison was making a big deal about her birthday earlier, and it's reasonable that everybody would know when it is. However, if Mac and Madison were sent home with each other's parents, then they were also assigned each other's birthdays, meaning that even though Madison was really born the day after Mac, Madison would celebrate her birthday the day before Mac celebrates hers.
Other mistake: Season 1, episode 19, Hot Dogs. After Veronica bugs the Spanish-speaking man who returned the dog in the park, a shot of her laptop is shown. It's the exact same shot as when Veronica was helping the valedictorian front-runner find out who was harassing her. There are books and even the girl she was helping in the shot, proving that the shot was not in Veronica's car, as seen in the very next subsequent shot.