Family Drama, Realism, Southern Gothic, Tragedy
Unfortunately, "Super Iconic Sex N' Madness Spectacular" isn't a viable genre. Dang.
A Streetcar Named Desire is actually realism of several different varieties. First you’ve got Magical Realism, which is a generally realistic setting with some odd fantasy thrown in. In this case, the fantasy enters the picture when the audience gets to see and hear some of Blanche’s imagined horrors: shadows on the wall, the eerie polka music overhead, the sounds of echoing voices. We can also call it Psychological Realism for these same reasons: at times it portrays reality as it exists in the mind, not as it exists objectively. Lastly there’s Social Realism, because of the play’s frank treatment of issues like immigration, class, gender roles, and power plays between women and men.
All that creepy shadows-on-the-wall voices-in-Blanche’s-head stuff that we talked about also explains the play’s categorization as Southern Gothic. Sure, the "supernatural" elements of the play turn out to be only in Blanche's imagination, but that’s the case in many horror movies. Oh, and Streetcar is a great portrait of social issues in New Orleans (ahem, the South) in the 1940's. But you knew that.
Lastly, because this is a play (a drama), it’s all expressed through dialogue and action—and okay, fine, through Williams’s indulgent stage directions. Family factors in big-time to all the dramatic goings-on, from Blanche and Stanley (brother and sister-in-law) to Stanley and Stella (husband and wife) to Stella and Blanche (sisters).