Building on recent literature on supervision practice that has turned away from previous efforts to construct typologies, and towards 'dialogic' models that emphasise iterative feedback processes between students and supervisors "in situ," this article examines how the curiosity of the supervisor expressed in supervision meetings can both model a relationship to scholarship and collegiality and support the development of confidence and self-trust in the doctoral candidate. Drawing on a qualitative study of video-recorded supervision meetings across multiple Australian universities, this article examines the entanglements of scholarly discourse, interpersonal conviviality, and curiosity within supervision relationships. To understand this, we adopt a 'post-critical' approach to doctoral training and borrow the concept of 'tacit knowledge' to consider the role of trust, conviviality, and informal 'know-how' in the development of formalised expertise. Analysis of exchanges within supervision meetings encourages the consideration of care as a relational structure linked to practices of curiosity and the sharing of tacit knowledge. We argue that although institutional pressures may continue to reshape doctoral candidatures in the neoliberal university, supervision meetings offer important sites for developing doctoral candidates' intellectual self-trust, including through the expression of curiosity by their supervisors.