Musgrave studied for three years at the University of Edinburgh, taking premedical courses; she also took music courses at the university and eventually received a Bachelor of Music degree (1950). From 1950 to 1954 she studied in Paris, chiefly with Nadia Boulanger (Boulanger, Nadia). In 1953 her first commission, Suite o' Bairnsangs (for voice and piano), was performed in Braemar, Scot., followed the next year by a Scottish BBC performance of Cantata for a Summer's Day. These and other early works were chiefly diatonic and suggestive of Scottish or medieval themes. Soon she turned to chromaticism and, later, serialism, producing the Piano Sonata (1956), String Quartet (1958), and other chamber works.

In the 1960s she continued to compose chamber works and vocal pieces but also turned to larger works, culminating in the three-act opera The Decision (first performed 1967), a drama on the ordeal of a trapped miner told in abstract instrumental terms. She continued to write operas, often on historical or classical themes, among them The Voice of Ariadne (1974), Harriet, the Woman Called Moses (1984), Simón Bolívar (1993), and Potalba (2003), set at the time of the Louisiana Purchase in the United States. Her ballets include Beauty and the Beast (1969) and Orfeo (1975). The dramatic themes carry through to abstract works: in the Clarinet Concerto (1968) the soloist moves around the stage to engage with different sections of the orchestra, and in the Horn Concerto (1971) the French horns stand in different parts of the concert hall. She added electronic sounds, often from prerecorded tapes, to much of her music in the 1970s and '80s.

Among Musgrave's later works are Narcissus for flute with digital delay (1987; also scored for clarinet, 1987), Three Women—Queen, Slave, Mistress for soprano and orchestra (1997), and Phoenix Rising for orchestra (1997).

In 1972 Musgrave moved to the United States, and in the 1970s she began conducting many of her works with orchestras in Scotland and the United States. She was a professor at Queens College of the City University of New York from 1987 to 2002, and for years she maintained a close relationship with the Virginia Opera in Norfolk, where several of her operas premiered in 1979 through 1995. Frequently asked to comment on being a woman composer, Musgrave said, “Yes, I am a woman; and I am a composer. But rarely at the same time.”