The ending of the short story is one where everything has changed.  Connie is almost transcendent in how her world has changed, and how everything is "round."  In this light, it is difficult to find a "flat" character.  One could say that the mother is flat because she does not change in how she perceives and critiques Connie.  The ending does change this a bit because Connie recognizes before Arnold takes her away that she will never see her mother again.  This particular revelation makes the mother's character a bit more round because the reader recognizes that on a level that might only be present on Connie's end, their relationship is complex.  I would say that June might be a character that is flat, to a great extent.  June is conventional and does exactly what is expected of her and is depicted in a manner that is flat and rather static.  Yet, this might be due to the fact that Connie sees her as this, someone incapable of complexity because she is following what prescribed notions of the good would dictate of her.  In the end, I think that Connie's father might be the most flat character because he exists, but is more concerned with working, reading the newspaper, and going to bed.  He is flat in his narrow state of being, one that does not include parenting and advising his daughter on how to make better choices, something that she recognizes too late at the end.