ER (1994)

Homeless for the Holidays - S3-E10

Plot hole: When Mark slips and falls in the staff lounge, he puts both hands on the table to catch himself - but we can see him (the actor) quite deliberately lower his head to one hand - this is supposedly where he cuts himself, but he actually hits his hand and not the table. This would not produce the deep cut he ended up with. Also, the head makes contact on the left of his forehead and he ends up with a cut on the right. And when we actually see the cut, it is vertical and logic tells us that it would have been impossible for him to have a vertical cut from the angle he fell.

The Long Way Around - S3-E15

Plot hole: Carol and Duncan escape out the back of the shop - but there are no police waiting for them, which makes no sense at all. In reality, with such a situation, the police would have been waiting at the back door. There seems to be no one around at all, as they were able to dash through an apartment. When they finally do run into a cop, he automatically assumes that Carol is an innocent bystander, and not one of the robbers. He couldn't have known this, and in reality, a policeman wouldn't assume it.