Stay Out Of The Basement (2) - S1-E13
Stupidity: When the real Dr. Brewer is found in the basement, he has duct tape over his mouth and needs his daughter to remove it for him. His hands are tied around the wrist, but not behind his back, so he could have easily removed the tape from his mouth himself long before he was found.
Stupidity: When Kellin is hit with the neurolytic emitter, it erases her memories of the outside world. However, the effects aren't immediate and take some time (enough to get her to sick bay and have time to talk to Chakotay). However, she spends her time telling Chakotay to help her remember their relationship, etc. when her memory is gone instead of just making a recording of herself describing the relationship and her feeling which can then be shown to her after her memory is gone. She might not believe the recording, but seeing herself discuss the situation would be better than just relying on Chakotay (who will soon be a stranger she never met) telling her something so unbelievable.
Stupidity: And the end, after sliding to another world, the group is caught on film by someone in her house. Quinn goes (illegally) into the home to get the tape. There is no point to this. Throughout the series, many people have seen them arrive, or leave, and usually when there's a mix up with one of their doubles, they spend so much time trying to convince people they are in fact from another world but can't prove it. Quinn doesn't even bother to see how long they're going to be on that world first.
A Rough Whimper of Insanity - S1-E7
Stupidity: The hacker's VR goggles are not plugged in to anything (despite having a wire for this purpose) when he and Tom sit down at the computer in the teacher's house.