Persistent homology is a powerful technique for probing and analyzing the shape of data with complex distributions and has been widely used in studies of global organization of data across spatial scales. Much richer information can be uncovered through local and regional topology; however, naive localization is prone to instabilities as localization can be extremely sensitive to sampling and noise. In this talk, I will present an approach to local homology that is provably robust and discuss how it may be used in shape analysis, particularly in situations where invariance under group actions needs to be taken into account. I will begin with the case of Euclidean data and then discuss an extension to data on other metric spaces. In this formulation, local homology across scales is viewed as a path of barcodes or persistence diagrams that is stable with respect to the Wasserstein distance. In addition to illustrations using synthetic data, I will present an application to quantitative trait loci analysis of tomato leaf shape, a collaboration with researchers at the Danforth Plant Research Center. Here, the primary goal is to discover interpretable associations between genotypes and complex phenotypes to elucidate the genetic basis of plant morphology.