Continuity mistake: Carrie says that she "didn't go" to college because Arthur felt that she would be abandoning him. He points out that her mother had just died. According to the episode "Class Struggle" (Season Two), Carrie actually went to college and then her mother died, causing her to drop out. In that episode, she wants to "go back" to college.
Home Cheapo - S9-E7
Continuity mistake: When Carrie finds out that Kelly and Deacon have afforded a lakeside house, she makes reference that Kelly doesn't even work. But in the previous episode ("Brace Yourself"), Kelly has a job. They even need to hire Spence to help out.
Suggested correction: What if she quit the job, her employment was terminated, or it was a temporary/seasonal position?

Party Favor - S2-E13
Continuity mistake: When Doug is mimicking the Japanese chef, the towel on the side moves positions without being touched. (00:01:00)

Time Share - S1-E22
Continuity mistake: When Doug Carrie and Ritchie are in the kitchen discussing how long Ritchie has to keep seeing Dorothy, Doug is making a snack involving some lettuce. The size, shape and position of this lettuce is constantly changing between subsequent shots. (00:15:20)

Continuity mistake: At the start when Arthur is ranting to Doug, the way Doug holds his coffee flask differs ever so slightly between shots. (00:07:00)

Continuity mistake: Doug comes out from the payphone and asks Carrie where did everyone go. When Carrie replies, the way she holds her glass changes between shots. (00:17:50)

Fixer Upper - S1-E12
Continuity mistake: When Doug is talking to Spencer in the subway, the people in the line behind him change instantly from 2 men to 1 man and 1 woman. (00:19:30)

Head First - S1-E6
Continuity mistake: When the gang walk into Coopers, a woman and man are talking outside on the street. When the camera changes to focus on Arthur a second man has appeared next to the other two and should have been visible in the first shot. (00:03:40)

Continuity mistake: When making a PB&J sandwich, the jar of peanut butter rotates by itself. Note the label. (00:09:30)

Continuity mistake: When Doug falls off his chair his right arm is on the floor in one shot and on the desk part of the chair in the next. (00:09:10)