She portrayed a gallery of character roles that extended from twentieth-century works by Leonard Bernstein, Benjamin Britten, Carlisle Floyd, Lee Hoiby, Arthur Honegger, Gian Carlo Menotti, and Douglas Moore, to the contralto heroines of Gilbert and Sullivan, and comic scene-stealers by Puccini, Mozart, and Donizetti.

In 1963 Greenspon joined the roster at the New York City Opera, beginning her tenure with the company portraying Olga Olsen in Kurt Weill's Street Scene.

Also in her repertoire were Geneviève in Debussy's Pelléas et Mélisande, Dame Quickly in Sarah Caldwell's production of Verdi's Falstaff, Frau von Luber in Weill's Der Silbersee, and the name part in Offenbach's La Grande-Duchesse de Gérolstein.

Costa-Greenspon took on leading roles for NYCO in the American premieres of Prokofiev’s The Fiery Angel (1965) and von Einem's Dantons Tod (1966), and the 1964 world-premiere of Hoiby's Natalia Petrovna (A Month in the Country).

In 1995, Costa-Greenspon made her only appearance at the Metropolitan Opera, as the Duchesse of Krakentorp, a speaking role, in La fille du régiment with June Anderson as Marie and Luciano Pavarotti as Tonio.