The parameters of source and strength are often presented as crucial for the semantic profile of modal verbs expressing root necessity such as ‘should’, ‘ought to’ and ‘be supposed to’. Their role, however, is hard to assess because of the lack of clear definitions. This article offers a new perspective on the nature of subjectivity and strength that is grounded in detailed, qualitative corpus analysis of sentences with ‘should’, ‘ought to’ and ‘be supposed to’. It operationalises and refines both parameters and applies them to an extensive corpus of 1200 Present-day English sentences with ‘should’, ‘ought to’ and ‘be supposed to’, which results in an updated and more accurate picture of the meaning distinctions between them. More generally, the study shows that an accurate interpretation of modal meanings is to a large extent dependent on the discourse context, and that a detailed corpus analysis is required to understand the complexity of the parameters interacting in the establishment of modal meanings in context.