What evidence of Modernism is apparent in Joyce's "Eveline"?
Concerning Joyce's "Eveline" and Modernism, one doesn't really search a Joyce story for evidence of Modernism. Joyce defines Modernism. What Joyce does in his stories came to be known as Modernism (along with what a few other artists do). Literature shouldn't be reduced to a checklist.
That said, one technique that Joyce uses in the story that is now considered an element of Modernism is the centering on the character Eveline's thoughts. Though the story, an early work by Joyce, is not pure stream-of-consciousness, it is about Eveline's thoughts and emotions. For Joyce, there is no absolute reality, only reality filtered through a character's point of view as revealed in her thoughts. The reader is told Eveline's thoughts as she decides to leave, prepares to leave, thinks about her leaving, and ultimately becomes paralyzed and doesn't leave.
Eveline also suffers from alienation, a theme typical of Modernist works. She is a female trapped in a male-dominated world, thrust into a role that should be reserved for a mother rather than a daughter and sister. She is stuck between what she feels her mother would want her to do, what society expects her to do, and what she as a person needs to do to achieve a life worth living.
Finally, Joyce uses a great deal of symbolism in the story. The character's reactions to these symbols serve as characterization of the character. Eveline, for instance, reacts to the house built by a man from Belfast (thus, a Protestant invader) as well as the voice of her mother repeating a nonsensical phrase.
James Joyce's "Eveline" evinces aspects of Modernism with a theme about social conditions, the internal monologue, and the open-ending. As one of the stories from The Dubliners, "Eveline" reflects the Irish social condition in which Catholicism dominates the actions and thoughts of individuals to the point of repression. Eveline passes up a romantic relationship for the promises she has made to her dying mother and to the Blessed Margaret Mary Alacoque, whose portrait hangs on the wall, blessing the house. Her indecisiveness that leads her to remain is indicated in her interior monologue in which she engages in self debate, asking herself if it is wise to have agreed to leave home; she wonders what "they will say of her in the Stores" and what it will be like to live in a distant country and "explore another life with Frank"; then, she hopes that Frank will "save her" from her abusive life at home with her father. Finally, however, on the day of her arranged departure, Eveline experiences the Joycean paralysis as she struggles with her sense of duty and desire for happiness. Yet, the ending is unresolved as "her eyes gave him [Frank] no sign of love or farewell or recognition," so Joyce leaves the open ending to symbolic interpretation by the reader, a certain Modernist technique.