For most complex traits, gene regulation is known to play a crucial mechanistic role as demonstrated by the consistent enrichment of expression quantitative trait loci (eQTLs) among trait-associated variants. Thus, understanding the genetic architecture of gene expression traits is key to elucidating the underlying mechanisms of complex traits. However, a systematic survey of the heritability and the distribution of effect sizes across all representative tissues in the human body has not been reported. Here we fill this gap through analyses of the RNA-seq data from a comprehensive set of tissue samples generated by the GTEx Project and the DGN whole blood cohort. We find that local h2 can be relatively well characterized with 50% of expressed genes showing significant h2 in DGN and 8-19% in GTEx. However, the current sample sizes (n < 362 in GTEx) only allow us to compute distal h2 for a handful of genes (3% in DGN and <1% in GTEx). Thus, we focus on local regulation. Bayesian Sparse Linear Mixed Model (BSLMM) analysis and the sparsity of optimal performing predictors provide compelling evidence that local architecture of gene expression traits is sparse rather than polygenic across DGN and all 40 GTEx tissues examined. To further delve into the tissue context specificity, we decompose the expression traits into cross-tissue and tissue-specific components. Heritability and sparsity estimates of these derived expression phenotypes show similar characteristics to the original traits. Consistent properties relative to prior GTEx multi-tissue analysis results suggest that these traits reflect the expected biology. Finally, we apply this knowledge to develop prediction models of gene expression traits for all tissues. The prediction models, heritability, and prediction performance R2 for original and decomposed expression phenotypes are made publicly available (https://github.com/hakyimlab/PrediXcan).