MOTIVATION: Clustering patient omic data is integral to developing precision medicine because it allows the identification of disease subtypes. A current major challenge is the integration multi-omic data to identify a shared structure and reduce noise. Cluster analysis is also increasingly applied on single-omic data, for example, in single cell RNA-seq analysis for clustering the transcriptomes of individual cells. This technology has clinical implications. Our motivation was therefore to develop a flexible and effective spectral clustering tool for both single and multi-omic data. RESULTS: We present Spectrum, a new spectral clustering method for complex omic data. Spectrum uses a self-tuning density-aware kernel we developed that enhances the similarity between points that share common nearest neighbours. It uses a tensor product graph data integration and diffusion procedure to reduce noise and reveal underlying structures. Spectrum contains a new method for finding the optimal number of clusters (K) involving eigenvector distribution analysis. Spectrum can automatically find K for both Gaussian and non-Gaussian structures. We demonstrate across 21 real expression datasets that Spectrum gives improved runtimes and better clustering results relative to other methods. AVAILABILITY AND IMPLEMENTATION: Spectrum is available as an R software package from CRAN https://cran.r-project.org/web/packages/Spectrum/index.html. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.