Amanda, originally from a genteel Southern family, regales her children frequently with tales of her idyllic youth and the scores of suitors who once pursued her. She is disappointed that Laura, who wears a brace on her leg and is painfully shy, does not attract any gentlemen callers. She enrolls Laura in a business college, hoping that she will make her own and the family’s fortune through a business career. Weeks later, however, Amanda discovers that Laura’s crippling shyness has led her to drop out of the class secretly and spend her days wandering the city alone. Amanda then decides that Laura’s last hope must lie in marriage and begins selling magazine subscriptions to earn the extra money she believes will help to attract suitors for Laura. Meanwhile, Tom, who loathes his warehouse job, finds escape in liquor, movies, and literature, much to his mother’s chagrin. During one of the frequent arguments between mother and son, Tom accidentally breaks several of the glass animal figurines that are Laura’s most prized possessions.
Amanda and Tom discuss Laura’s prospects, and Amanda asks Tom to keep an eye out for potential suitors at the warehouse. Tom selects Jim O’Connor, a casual friend, and invites him to dinner. Amanda quizzes Tom about Jim and is delighted to learn that he is a driven young man with his mind set on career advancement. She prepares an elaborate dinner and insists that Laura wear a new dress. At the last minute, Laura learns the name of her caller; as it turns out, she had a devastating crush on Jim in high school. When Jim arrives, Laura answers the door, on Amanda’s orders, and then quickly disappears, leaving Tom and Jim alone. Tom confides to Jim that he has used the money for his family’s electric bill to join the merchant marine and plans to leave his job and family in search of adventure. Laura refuses to eat dinner with the others, feigning illness. Amanda, wearing an ostentatious dress from her glamorous youth, talks vivaciously with Jim throughout the meal.
As dinner is ending, the lights go out as a consequence of the unpaid electric bill. The characters light candles, and Amanda encourages Jim to entertain Laura in the living room while she and Tom clean up. Laura is at first paralyzed by Jim’s presence, but his warm and open behavior soon draws her out of her shell. She confesses that she knew and liked him in high school but was too shy to approach him. They continue talking, and Laura reminds him of the nickname he had given her: “Blue Roses,” an accidental corruption of pleurosis, an illness Laura had in high school. He reproaches her for her shyness and low self-esteem but praises her uniqueness. Laura then ventures to show him her favorite glass animal, a unicorn. Jim dances with her, but in the process, he accidentally knocks over the unicorn, breaking off its horn. Laura is forgiving, noting that now the unicorn is a normal horse. Jim then kisses her, but he quickly draws back and apologizes, explaining that he was carried away by the moment and that he actually has a serious girlfriend. Resigned, Laura offers him the broken unicorn as a souvenir.
Amanda enters the living room, full of good cheer. Jim hastily explains that he must leave because of an appointment with his fiancée. Amanda sees him off warmly but, after he is gone, turns on Tom, who had not known that Jim was engaged. Amanda accuses Tom of being an inattentive, selfish dreamer and then throws herself into comforting Laura. From the fire escape outside of their apartment, Tom watches the two women and explains that, not long after Jim’s visit, he gets fired from his job and leaves Amanda and Laura behind. Years later, though he travels far, he finds that he is unable to leave behind guilty memories of Laura.