Jones is nominated for "Driving Miss Daisy," Law for "Anna Christie" and Spacey for "Richard III," alongside Benedict Cumberbatch for "Frankenstein," James Corden for "One Man, Two Guvnors" and David Tennant for "Much Ado About Nothing."
The prizes, run by theater website whatonstage.com, are decided by public vote.
Best actress contenders announced Friday include Vanessa Redgrave for "Driving Miss Daisy" and Kristin Scott Thomas for "Betrayal."
In the musical categories, there are multiple nominations for the movie-inspired romance "Ghost" and Roald Dahl-based "Matilda."
Winners will be announced Feb. 19.