The concept of urban sustainability is growing in urgency and salience for local urban governance, and indicator-based assessments represent a popular means for its operationalization. While much effort has been spent developing the technical aspects of these assessments, less attention has been given to research concerning their potential for influencing real-world urban governance processes. To address this issue, we put forward an assessment approach that systematically embeds the assessed indicators within their social and institutional contexts, thereby aiming to enhance the relevance of the assessment for local governance. Specifically, the contextual embedding is achieved through the analysis of ongoing controversies related to the assessed problem. We apply the approach to an assessment of the City of Geneva’s (Switzerland) housing system. As the case study demonstrates, the proposed assessment approach can elucidate a richer picture of the challenges identified in the assessment than a typical quantitative-only analysis of indicators. Therefore, it offers more complete support to local governance stakeholders for learning about and acting upon the problem under assessment. Overall, the work reported in this article aims to contribute to a productive alliance between sustainability assessment methodologies and urban governance stakeholders, thereby leading to more informed steering of cities towards sustainability.