Continuity mistake: None of the Sheffield children are aged properly over the course of the series. It's established that Grace is 6 years old in season 1, but two years later in season 3 she's 10. Maggie is 18 at the end of season 5 but turns 21 years old in early season 6. And finally, Brighton is in seventh grade in season 2 and somehow is graduating high school at the end of season 6 (when he should be only a junior, and nothing about Brighton's character would suggest he's capable of skipping a grade).
Continuity mistake: In the opening sequence, Fran arrives at the Scheffield house with a case of make-up. When we see a shot of the makeup head on, one can see two lipsticks at the far left corner of the case. When we see a shot of the case from behind, the lipstick is now at the right corner of the case.
Other mistake: Pretending to be Max's assistant, Fran dials a number for Bernadette Peters. The conversation is on speaker phone. The assistant says "one moment please." Niles and Fran continue talking and Maxwell comes in. Fran and Niles sheepishly hurry out of the room. Notice how, this whole time, Fran never touched the phone again to turn the phone off. So Max would hear Bernadette Peters and actually figure out Niles and Fran's plan before the episode's even halfway over.