The concept of sustainability is growing in importance 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 their potential for influencing real-world governance processes. To address this issue, we put forward an assessment approach that systematically embeds the assessed indicators within their sociopolitical and institutional contexts, thereby aiming to enhance the informational value of the assessment for local governance. We apply the approach to the assessment of the City of Geneva's (Switzerland) housing system, for which an assessment model is first developed, covering 13 goals and 26 indicators. The indicators reveal the most critical issues of Geneva's housing system, which include energy performance, availability and affordability, and certain aspects of the urban environment (noise, moderation of traffic, green areas, and mobility). We then deepen the analysis by connecting the indicators to ongoing controversies around the housing system, and to the stakeholders and institutions that these controversies touch upon. 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, our work 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.