Library of Congress Classification is not set up well for playscripts, but if you ever want to explore the stacks the old-fashioned way, you totally can!
But remember -- all literature is arranged by author last name, and plays will be mixed in with any other lit written by that author (poetry, novels, etc.), so browsing for plays takes fortitude of mind and body. Collections and Anthologies are a good place to start.
PS625 = Selections, Anthologies, Collections of plays
PS700 = 17th & 18th Century Plays (Colonial Period), in order by author last name
PS991 = 19th Century Plays, in order of last name author last name
PS3500 = Plays by authors who began writing between 1900-1960, in order by author last name
PS3550 = Plays by authors who began writing between 1960-2000, in order by author last name
PS3600 = Plays by authors who began writing between 2001-present, in order by author last name
PR1261.A --> .Z= Medieval Mystery Plays and Play Cycles
PR2411 = English Renaissance Plays (1500-1640) [Includes Shakespeare]
PR3291 = 17th & 18th Century Plays (1640-1770)
PR3991 = 19th Century Plays
PR6000 = Plays by authors who began writing between 1900-1960, in order by author last name
PR6050 = Plays by authors who began writing between 1960-2000, in order by author last name
PS6100 = Plays by authors who began writing between 2001-present, in order by author last name