Lady Macbeth of the Mtsensk District
The most popular operas by composers from the 20th century (i.e., by composers born after 1900) were "The Turn of the Screw" by Benjamin Britten and "Lady Macbeth of the Mtsensk District" (1934, Dmitri Shostakovich).  The most popular works by composers alive on 31 July 2012 were the monodrama "The Diary of Anne Frank" (composed 1968, Grigory Frid), and the opera "Dead Man Walking" (2000, Jake Heggie).