Shifts in data distribution can substantially harm the performance of clinical AI models. Hence, various methods have been developed to detect the presence of such shifts at deployment time. However, root causes of dataset shifts are varied, and the choice of shift mitigation strategies is highly dependent on the precise type of shift encountered at test time. As such, detecting test-time dataset shift is not sufficient: precisely identifying which type of shift has occurred is critical. In this work, we propose the first unsupervised dataset shift identification framework, effectively distinguishing between prevalence shift (caused by a change in the label distribution), covariate shift (caused by a change in input characteristics) and mixed shifts (simultaneous prevalence and covariate shifts). We discuss the importance of self-supervised encoders for detecting subtle covariate shifts and propose a novel shift detector leveraging both self-supervised encoders and task model outputs for improved shift detection. We report promising results for the proposed shift identification framework across three different imaging modalities (chest radiography, digital mammography, and retinal fundus images) on five types of real-world dataset shifts, using four large publicly available datasets.