Exploratory spatial data analysis (ESDA) can be a powerful approach to understanding single-cell genomics datasets, but it is not yet part of standard data analysis workflows. In particular, geospatial analyses, which have been developed and refined for decades, have yet to be fully adapted and applied to spatial single-cell analysis. We introduce the Voyager platform, which systematically brings the geospatial ESDA tradition to (spatial) -omics, with local, bivariate, and multivariate spatial methods not yet commonly applied to spatial -omics, united by a uniform user interface. Using Voyager, we showcase biological insights that can be derived with its methods, such as biologically relevant negative spatial autocorrelation. Underlying Voyager is the SpatialFeatureExperiment data structure, which combines Simple Feature with SingleCellExperiment and AnnData to represent and operate on geometries bundled with gene expression data. Voyager has comprehensive tutorials demonstrating ESDA built on GitHub Actions to ensure reproducibility and scalability, using data from popular commercial technologies. Voyager is implemented in both R/Bioconductor and Python/PyPI, and features compatibility tests to ensure that both implementations return consistent results.