Mobile applications for digital contact tracing have been developed and introduced around the world in response to the COVID-19 pandemic. Proposed as a tool to support ‘traditional’ forms of contact-tracing carried out to monitor contagion, these apps have triggered an intense debate with respect to their legal and ethical permissibility, social desirability and general feasibility. Based on a large-scale study including qualitative data from 349 interviews conducted in nine European countries (Austria, Belgium, France, Germany, Ireland, Italy, the Netherlands, German-speaking Switzerland, the United Kingdom), this paper shows that the binary framing often found in surveys and polls, which contrasts privacy concerns with the usefulness of these interventions for public health, does not capture the depth, breadth, and nuances of people’s positions towards COVID-19 contact-tracing apps. The paper provides a detailed account of how people arrive at certain normative positions by analysing the argumentative patterns, tropes and (moral) repertoires underpinning people’s perspectives on digital contact-tracing. Specifically, we identified a spectrum comprising five normative positions towards the use of COVID-19 contact-tracing apps: opposition, scepticism of feasibility, pondered deliberation, resignation, and support. We describe these stances and analyse the diversity of assumptions and values that underlie the normative orientations of our interviewees. We conclude by arguing that policy attempts to develop and implement these and other digital responses to the pandemic should move beyond the reiteration of binary framings, and instead cater to the variety of values, concerns and expectations that citizens voice in discussions about these types of public health interventions.