Genetic correlation (rg) analysis is commonly used to identify traits that may have a shared genetic basis. Traditionally, rg is studied on a global scale, considering only the average of the shared signal across the genome; though this approach may fail to detect scenarios where the rg is confined to particular genomic regions, or show opposing directions at different loci. Tools dedicated to local rg analysis have started to emerge, but are currently restricted to analysis of two phenotypes. For this reason, we have developed LAVA, an integrated framework for local rg analysis which, in addition to testing the standard bivariate local rg’s between two traits, can evaluate the local heritability for all traits of interest, and analyse conditional genetic relations between several traits using partial correlation or multiple regression. Applied to 20 behavioural and health phenotypes, we show considerable heterogeneity in the bivariate local rg’s across the genome, which is often masked by the global rg patterns, and demonstrate how our conditional approaches can elucidate more complex, multivariate genetic relations between traits.