Most pieces of musical theatre contain what is called book—that is, spoken dialogue scenes in between each of the songs. However, some decide to go all the way and ditch spoken word completely (or almost completely) for sung word. Hence, this trope. More conversational and utilitarian exchanges are commonly made with use of recitative, in which the delivery of the singing is meant to mimic regular speech.
It is worth noting that most of these shows do have brief bits of spoken dialogue. In Evita
, for instance, Che has two short monologues between and within songs. However, in order to qualify there should not be any kind of scenes told through dialogue. Hence, Sweeney Todd: The Demon Barber of Fleet Street
does not qualify because even though it is very predominantly sung, there are multiple scenes of spoken dialogue.