The sell-out Barbican show has nine nominations, including best actor for the Sherlock star.
Musicals Kinky Boots and In The Heights also have nine nominations each.
The winners, chosen by public vote, will be announced at an awards concert on 21 February 2016 at the Prince of Wales Theatre.
Hamlet's nominations include supporting actor nods for Ciaran Hinds, Kobna Holdbrook-Smith, Sian Brooke and Anastasia Hille, while Lyndsey Turner is shortlisted for best director.
The production made headlines this summer after it shifted the position of Hamlet's famous "to be or not to be" soliloquy.
Cumberbatch had also made a plea to fans during early performances not to film his performance.
The 39-year-old actor faces competition in the best actor category from James McAvoy (The Ruling Class), Bradley Cooper (The Elephant Man), Mark Rylance (Farinelli and the King) and Alex Hassell (Henry V).
Nicole Kidman is nominated for best actress in a play for her part as a pioneering scientist in Photograph 51.
She is up against Rosalie Craig (As You Like It), Denise Gough (People, Places, Things), Harriet Walter (Death Of A Salesman) and Lia Williams (Oresteia).
Photograph 51 and Farinelli and the King are both nominated for best new play, along with Oppenheimer, People, Places and Things and Hangmen.
The recently-opened Elf the Musical is up for best new musical category in competition with Kinky Boots, Bend It Like Beckham, Beautiful and In the Heights.
THE 16TH ANNUAL WHATSONSTAGE AWARDS NOMINEES IN FULL:
Best Actor in a Play:
Best Actress in a Play:
Best Actor in a Musical:
Best Actress in a Musical:
Best Supporting Actor in a Play:
Best Supporting Actress in a Play:
Best Supporting Actor in a Musical:
Best Supporting Actress in a Musical:
Best New Play:
Best New Musical:
Best Play Revival:
Best Musical Revival:
Best Direction:
Best Choreography:
Best Set Design:
Best Lighting Design:
Best Off-West End Production:
Best Regional Production:
Best West End Show:
The blockbuster production of Hamlet, starring Benedict Cumberbatch, is the most nominated play at the WhatsOnStage awards.