This is a somewhat difficult question to answer since there are virtually thousands of short stories that could be said to contain the theme of the American Dream! However, to help you I will mention a few authors and titles that fit this description.
The American Dream is a somewhat complex theme in that it may be addressed via the exploration of many sub-themes, such as provincialism, social class, colonialism, and gender (for example, the role of women in the home is related to the classic picture that illustrates the American Dream). The term was first coined by historian James Truslow Adams in 1931 to describe the expectations and aspirations of Americans at the dawn of a very difficult and turbulent area, the Great Depression. While the concept was meant to be hopeful and aspirational, in more recent years the term has come to be viewed with irony and disdain by many cultural critics.
The economy began to boom again just after the Second World War. In the post-war years (1950s) the rapid expansion of cities into suburbs was considered one way the American Dream could be fulfilled, since the suburbs at the time represented a quiet, cleaner, and healthier life than that found in the cities. Over time, however, suburban life came to be seen as stifling and shallow, and this theme is written about by many authors.
One author who wrote extensively about life in suburban America, whose work could be said to examine the theme of the American Dream at times, is John Updike. His short story "A & P" is found in many anthologies, and is a coming-of-age tale about a young man who works in a grocery store and a fateful day when he decides to change his life. His decision is catalyzed by the arrival of three teenage girls who come to the store and an incident that takes place involving one of them. The story considers the awareness that the young man has of the importance of work, and there are issues of social class and suburban snobbery that occur as well. Updike wrote many other short stories and novels about suburban life, including a best-selling series of four novels about a man named Harry whose nickname, Rabbit, occurs in the title of each one.
Another American author who wrote about the stifling quality of suburban life was John Cheever, whose stories often explored darker themes such as infidelity and alcoholism as well as mid-life crisis, particularly among men.
Author Pamela Zoline is an experimental writer whose short story collection The Heat Death of the Universe contains several stories on this theme as well, and the title story in particular explores the daily life of a woman who believes her suburban existence as a housewife and mother is slowly driving her insane.