Character mistake: At one point in the show, Roger throws a bucket of water (in disguise as Scotch Bingeington) on Francine claiming "Wet T-Shirt Contest!" Francine looks disgusted when Roger states "Eh, give this one a four" to the guy judging the contest, then says to Francine "Don't feel bad, you've had three kids." Francine replies "I've had two." and Roger gasps and puts a consoling hand on Francine's shoulder. This is wrong because Francine has birthed four children. She previously gave birth to Bailey (Hayley's twin), and Greg and Terry's baby in Season 3 Episode 7 "Surro-Gate." (00:14:15)
Character mistake: When Jet tells Katara and Aang to waterbend the water from underground in the geysers, Katara states that she has never waterbent on water she couldn't see before. However, six episodes prior to this, in 'The Warriors of Kyoshi', she waterbends the water out from Aang's lungs to save his life.
Suggested correction: That was a moment of desperation on her part, and she probably doesn't think of it/think she can do it again without the same desperate impetus, Furthermore, the water in Aang was only a small amount and very close anyway.
Regardless of if she did it in desperation or not, she did use bending on water she didn't see. The mistake isn't suggesting anything about Katara's ability to waterbend unseen water, the mistake is only in regards to the statement Katara made about never using bending on water she can't see. And for Katara to forget the one time she used bending on water she didn't see would qualify as a character mistake as well. Now if she made the statement "I don't think I can do this" or "I've never used bending on underground water", etc, that would not be a character mistake.
Takishîdo wa kaze ni mau - S1-E1
Character mistake: During Lucky's initial display of power at the gates of the settlement, he fires a cannon shot at the building and shouts (in the English dub) "How do you like that? Bullseye." There's a view of the building and you can read the horseshoe sign; "Evergreen SHOPING Mall." (00:07:15)