K-means, one of the simplest clustering algorithms, is ubiquitous in every scientific field. Its cost function supports many possible clustering solutions, and location of low-valued clustering solutions can be challenging. Hence, the topography of the cost function surface is crucial to understanding K-means performance. We present the application of energy landscape theory to the K-means cost function to elucidate its topography, which we deduce from K-means landscapes described in terms minima and transition states. We analyse K-means landscapes for Fisher’s Iris dataset, the glass identification dataset, and many variations in which we alter their properties. For K-means the number of clusters must be prespecified, and we consider the effect of that choice on the Iris landscapes. K-means landscapes are also constructed for the glass and Iris datasets with varying numbers of outliers, and different feature standardisations. Both the removal of outliers and standardisation are common procedures during dataset preprocessing. Moreover, we systematically change the cluster overlap and cluster populations for the two datasets, both of which are crucial to K-means accuracy. The K-means landscapes for all these modified datasets allows us to understand the effect of the most important dataset features on cost function topography. In all cases we observe that the cost function topography changes independently of K-means accuracy; a dataset modification that increases clustering accuracy can also make the cost function surface harder to explore. Therefore, the maximum accuracy is not always sufficient to understand K-means success, and we must also consider the feasibility of obtaining these accurate solutions. The K-means landscapes allow us to address the second of these considerations. For most dataset properties the resulting topography changes are largely systematic, and the concepts can be applied to novel datasets to predict K-means performance, highlight possible problems, and select a sampling strategy to improve future applications.