The show about two missionaries who find more than they bargained for in Africa, is up for original score, best musical and lead actor in a musical.
Al Pacino is up for best actor in a play for The Merchant of Venice, as is UK actor Mark Rylance for Jerusalem.
The stage awards ceremony takes place in New York on 12 June.
The Scottsboro Boys, based on the 1930s case in which nine black men were unjustly accused of attacking two white women on an Alabama train, picked up 12 nominations.
It is up for best musical, book of a musical, original score as well as a leading actor and two featured actor nods.
Cole Porter’s Anything Goes gained nine nominations, including best revival and best actress for Sutton Foster.
While How to Succeed in Business Without Really Trying garnered eight, although its star – Harry Potter’s Daniel Radcliffe – was not recognised.
Veteran actress Vanessa Redgrave earned a best actress in a play nod for her role in Driving Miss Daisy.
She will compete against Frances McDormand for Good People, Nina Arianda for Born Yesterday, Lily Rabe for The Merchant of Venice and Hannah Yelland for Brief Encounter.
There was also a nomination for Pirates of the Caribbean star Mackenzie Crook for his performance in Jez Butterworth’s Jerusalem.
And Absolutely Fabulous star Joanna Lumley also received a best supporting actress in a featured role nod for her performance in La Bete.
Ocean’s Thirteen star Ellen Barkin for The Normal Heart and Sopranos actress Edie Falco for The House of Blue Leaves are also on the shortlist.
