Carrie's handbag switches shoulders when she talks to Doug in the kitchen.
Arthur pours salad dressing on his salad and then hands the bottle back to Doug. The camera then switches to Doug's face. When the camera switches back to Arthur, Arthur once again pours salad dressing on his salad and once again hands the bottle back to Doug. This occurs at the 10:34 minute mark in this episode.
When Arthur comes into the kitchen while Doug and Deacon are having breakfast, he walks through an open door with his blue cardigan and shoes hanging from his right arm. After cutting to Doug and back to Arthur, the door is closed and the cardigan and shoes have switched arms.
When Carrie picks up the stack of papers from the printer, they are (more or less) lined up. In the next shot, one paper sticks out from the others. Then she gathers several papers in her left hand, and in the next shot her left hand is empty.
When Doug is listening to the baseball game in bed, the announcer says that Mike Piazza of the Mets hits a home run off of Greg Maddux of the Braves to win the game in walk off fashion it is assumed in the bottom of the ninth by a score of 9-8. In the real world, Maddux, who was a starter, would not be in a game that late having allowed that many runs.
(at around 25 mins) When Arthur approaches Doug in the living room, looking over Arthur's shoulder on the fridge you will see the shadow of the camera adjusting height.