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...so it might be best to start with Collections and Anthologies...
American Drama
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
British Drama
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