Graphene, a Dirac semimetal, exhibits the simplest lattice configuration and band structure in the world of two-dimensional materials. Due to its remarkable brevity and tunability, graphene becomes an ideal platform for studying the fundamental physics arising from the linear dispersion around the Dirac point, as well as for exploring symmetry-breaking orders in the flat band through playing with various artificial structures and external fields. In this review, we provide an overview of the nanoscale graphene model structures such as defects, quantum dots, strains, and superlattices in scanning tunneling microscopy measurements. Utilizing nanostructures in diverse dimensions, we present some behaviors of electrons near singularities of density of states from the perspective of scanning tunneling microscopy.