I am sort of surprised that no one has mentioned Ayn Rand. Atlas Shrugged and The Fountainhead as well as We The Living and Anthem all deal with universal social issues of class struggle, altruism and the virtue of selfishness. Check her out. Atlas Shrugged is a good novel right now because of the recently released movie based on the novel.
I you want to explore the problems of race and society, I would suggest Richard Wright's Native Son and Ralph Ellison's Invisible Man. Both of these novels deal with one individual who, in different ways, succumb to society's view of their race. The first novel deals with a black youth who violently kills a white woman and his flight from justice. The second novel deals with a man who attempts to blend into white society. Both are more appropriate for high school, and Wright's novel is a bit graphic for some readers. Check your school's policy!
"Social issues" covers a pretty broad collection of things for a text to potentially approach. As stated by a previous poster, any truly worthy piece of literature will approach social issues in one way or another. In order to get the most out of your task, I'd suggest narrowing down what types of issues you are hoping to illuminate through literature. Are they tied to a certain time or place? Your analytical context can change greatly depending on the location and historical period that the text is situated in. Another thing to consider: are the issues you're seeking to explore more about money (ex. poverty, social stratification), gender (ex. sexism, LGBT issues), environment, war, race, or politics? Depending on your answer, seek out some of the canonical texts approached by the schools of literary theory that specialize in these issues. For example, if gender inequality is the issue you'd prefer to examine, try an author frequently referenced in feminist criticism, such as Virginia Woolf, Adrienne Rich, or Toni Morrison. If race is the issue to tackle, go for something Post-Colonial (ex. Louise Erdrich, Chinua Achebe, or Salman Rushdie). These categories are already there to explore, so why not let a critical angle determine your direction?
I am going to have to echo some of the previous editors and say that Charles Dickens screamed to me from the grave when I read your discussion post. (Okay, not literally. Ha!) I consider Dickens to be the most profound writer in regards to social issues. However, it's important to note that Charles Dickens was only considering the specific social issues in his own, British historical context: the industrial revolution in England. It's also important to note that some of his works would be better to discuss social issues than others. This is why I couldn't ignore your post without putting in another vote for Oliver Twist. Could there be a better social issue to discuss than the intense pain of the working class?
In the world of classic novels, Harper Lee deals with lots of social issues such as race in To Kill a Mockingbird. The same is true of Fitzgerald's The Great Gatsby regarding social class and money. William Golding's Lord of the Flies is a chilling look at society without the rule of law.
Really, as someone mentioned above, any classic novel deals with how characters deal with personal issues (most of which are ultimately also social) or social issues. The key, then, is to search out the great novels.
If you're looking for more modern author's/social problems, you might want to check out Mary Pipher and her book The Middle of Everywhere. It deals with refugees and immigrants in America. If you were looking for other social problems, you might to check out Ernest Gaines (A Lesson Before Dying) or Johnathan Swift (A Modest Proposal). As far as a classic author with many books that deal with social issues, I would certainly agree with Charles Dickens.
Historically, to support #4, Dickens is famous for bringing the suffering of the working classes in England to the attention of a middle and upper-class readership. Novels such as Oliver Twist, Hard Times and Bleak House really served to capture the sympathies of his readership and in turn instigated social changes for the benefit of the working classes.
In the same vein as Toni Morrison, Zora Neale Hurston wrote about the search for self by women as did Kate Chopin and Charlotte Perkins Gilman during the Victorian period. And, certainly the greats such as Charles Dickens and Victor Hugo delved into the societal prisons and injustices of their times.
Toni Morrison writes eloquently about the life and blacks in the white dominated society. Her characters are all faced with very difficult personal situations that are a result of their personal as well as their social situation. Sherman Alexie writes about Native Americans and their lives on the reservations. So does Leslie Marmon Silko. Going in a different direction, Jane Austen wrote about the social situation of women in English society in the early 1800's. Tim O'Brien writes several books that explore the social problem of Vietnam War. A list could go on that has on it books from every generation and social/political situation in history.
This depends a lot on what level you are looking at. At a young teen level, S.E. Hinton is one who dealt with social problems and issues of importance to teens. She dealt with issues of alienation among teens and of violence that alienated teens engage in.
On a higher level, Huxley's Brave New World and Orwell's 1984 deal with issues of totalitarianism.
But really, practically every author of serious literature is dealing with social problems and issues, right?