At the ice rink, the only couple that includes a skating instructor are dressed slightly differently than Liane and the instructor. He appears to have a black hat on and she a large black bow on her hat. These items have disappeared when we see a close up of them skating.
When Gaston comes to talk with Grandmother, he sits down and unbuttons his jacket. The camera turns to her, and back, he stands up, his jacket has magically buttoned itself, and he once again unbuttons it!
In the scene where Alicia is teaching Gigi about the jewels, in a shot, Alicia is facing Gigi, having her hands pointing up and Gigi is facing her, listening. In the very next shot, Alicia is playing with a string of pearls, facing away from Gigi. Gigi is sitting beside her but also not facing her but still looking at her.
After announcing that the year is 1900, Honore evades the question of his age by saying he wasn't born "in this century". 1900 was the last year of the 19th century, not the first year of the 20th century, so he was in fact born in the present century.
During her visit with Aunt Alicia, Gigi says she has no friends and that she's always on her own. However, in the beginning of the movie, we see Gigi playing with several other schoolgirls, which would seem to indicate she does have friends.
During Gaston's song by the pond thinking of Gigi, there is a fence in the pond forcing the swans to stay in close background. The swans, obviously confused yet undeterred, keep swimming into the fence attempting to get to the swan in deep background on the other side of the pond.
The café where Honore (Maurice Chevalier) sings "I'm Glad I'm Not Young Anymore" is clearly a studio set. As he leaves the café at the conclusion of the number, a horse and carriage pull up on the street in the background, and their shadows are seen on the painted backdrop of the buildings at the back of the set.
In the early scene in the park, the principal actor casts a shadow backwards (away from the viewer), while the carriages in the background cast shadows forwards (towards the viewer), revealing the production lighting.
The goof item below may give away important plot points.
When Gaston asks Mamita for Gigi's hand in marriage, it is a medium shot with Gigi in the background, raising her hand to her bosom. In the closeup of Gigi immediately following, she puts her hand up again.