We introduce a new technique to visualize complex flowing phenomena by using concepts from shape analysis. Our approach uses techniques that examine the intrinsic geometry of manifolds through their heat kernel, to obtain representations of such manifolds that are isometry-invariant and multi-scale. These representations permit us to compute heat kernel signatures of each point on that manifold, and we can use these signatures as features for classification and segmentation that identify points that have similar structural properties. Our approach adapts heat kernel signatures to unsteady flows by formulating a notion of shape where pathlines are observations of a manifold living in a high-dimensional space. We use this space to compute and visualize heat kernel signatures associated with each pathline. Besides being able to capture the structural features of a pathline, heat kernel signatures allow the comparison of pathlines from different flow datasets through a shape matching pipeline. We demonstrate the analytic power of heat kernel signatures by comparing both (1) different timesteps from the same unsteady flow as well as (2) flow datasets taken from ensemble simulations with varying simulation parameters. Our analysis only requires the pathlines themselves, and thus it does not utilize the underlying vector field directly. We make minimal assumptions on the pathlines: while we assume they are sampled from a continuous, unsteady flow, our computations can tolerate pathlines that have varying density and potential unknown boundaries. We evaluate our approach through visualizations of a variety of two-dimensional unsteady flows.