SARS-CoV-2 is a positive-sense single-stranded RNA virus that has exploded throughout the global human population. This pandemic coronavirus strain has taken scientists and public health researchers by surprise and knowledge of its basic biology (e.g. structure/function relationships in its genomic, messenger and template RNAs) and modes for therapeutic intervention lag behind that of other human pathogens. In this report we used a recently-developed bioinformatics approach, ScanFold, to deduce the RNA structural landscape of the SARS-CoV-2 transcriptome. We recapitulate known elements of RNA structure and provide a model for the folding of an essential frameshift signal. Our results find that the SARS-CoV-2 is greatly enriched in unusually stable and likely evolutionarily ordered RNA structure, which provides a huge reservoir of potential drug targets for RNA-binding small molecules. Our results also predict regions that are accessible for intermolecular interactions, which can aid in the design of antisense therapeutics. All results are made available via a public database (the RNAStructuromeDB) where they may hopefully drive drug discovery efforts to inhibit SARS-CoV-2 pathogenesis.