Deliberate mistake: Every time the investigators deal with IP-addresses, the addresses on display are impossible. Each of the four parts of an IP-address has to be between 0 and 255. As they do have to use IP-addresses some time, they could use addresses starting with 10. Those would be real addresses although not used as an official IP-address. This isn't the same as phone numbers using 555 - any IP address over 255 just wouldn't work. It would be like mentioning a phone number which uses the symbol for pi.
Deliberate mistake: In Series 1 episode 1 Nick can use a laptop fluently, yet in series 1 episode 2, Nick has to ask where the "." button is.
Episode 306 - S3-E6
Deliberate mistake: In the locker scene where Brian confronts Ben about his steroid use, there are no obvious objects on the lockers (such as slats) that would make the cut and bruise on Brian's back. Brian is not tall enough to have been cut by the top of the locker and Ben didn't pick up in his tirade. (00:25:25)
Deliberate mistake: Towards the end, Logan is thinking about killing himself and the water drips from the ceiling. He is on the top floor, so nobody would be living above him.
Snow Place to Go/Dunce Upon a Time/Gems the Breaks - S2-E4
Deliberate mistake: Dunce Upon a Time: When Lumpy falls off the beanstalk and hits the ground, his left antler is missing. Probably done deliberately so that his head could fall flat on the ground. (00:14:03)