Let us first define the traits of a flat character in order to appropriately attribute them.
A flat character is one which does not go through any significant changes as a result of the sequence of events that occur in the novel. This is a character that serves as a support system to the main character who is usually round, or changing, also known as "dynamic". Moreover, a flat character is two-dimensional which means that the reader does not learn a lot of information about their inner thoughts and emotions; they are an active part of the plot, but are not affected by it. Therefore, we could conclude that the character of Phoebe Caulfield is a flat character.
Phoebe, who is much more intellectually and socially mature than her older brother Holden, preserves these very traits all throughout Holden's narrative. This is the first fact that would define her as a flat character. The second fact is that she serves as an anchor of support for Holden. So important is her presence in Holden's life that it is Phoebe who inspires his wish to become the catcher in the rye. Phoebe is Holden's go-to support mechanism and the mutual love that exists between Phoebe and Holden does not change either.
As a flat character, Phoebe is also two-dimensional. We do learn some information about her, but hardly any really intimate facts that reveal her differently to the reader. She remains the same Phoebe in the eyes of Holden, and in the reader's own eyes.
Lastly, but not less importantly, the events in Holden's life do not directly affect Phoebe. She remains on the sidelines performing her role as Holden's support system. The fact that she is unaffected confirms that she is a flat character. Had she undergone any significant changes as a result of Holden's problems, then she would have been a round or dynamic character.