Treatment of renal calculi is highly dependent on the chemical composition of the stone in question, which is difficult to determine using standard imaging techniques. The objective of this study is to evaluate the potential of scatter-sensitive X-ray dark-field radiography to differentiate between the most common types of kidney stones in clinical practice. Here, we examine the absorption-to-scattering ratio of 118 extracted kidney stones with a laboratory Talbot-Lau Interferometer. Depending on their chemical composition, microscopic growth structure and morphology the various types of kidney stones show strongly varying, partially opposite contrasts in absorption and dark-field imaging. By assessing the microscopic calculi morphology with high resolution micro-computed tomography measurements, we illustrate the dependence of dark-field signal strength on the respective stone type. Finally, we utilize X-ray dark-field radiography as a non-invasive, highly sensitive (100%) and specific (97%) tool for the differentiation of calcium oxalate, uric acid and mixed types of stones, while additionally improving the detectability of radio-lucent calculi. We prove clinical feasibility of the here proposed method by accurately classifying renal stones, embedded within a fresh pig kidney, using dose-compatible measurements and a quick and simple visual inspection.