Single molecular systems are typical quantum confinement systems, which have rich electronic states, photon states and spin states due to their discrete energy levels, localized orbitals and diverse chemical structures. The states determined by quantum mechanics in these molecular systems make it possible to serve as great physical entities for future quantum information technology. The detection and manipulation of quantum states on a single molecule scale are beneficial to the bottom-up construction of quantum devices. Owing to the highly limited spatial localization of single molecular systems, it is difficult to accurately address and manipulate them with conventional macroscopic characterization methods. Scanning tunneling microscope (STM) is such a powerful tool that it can achieve high-resolution real-space imaging as well as spectroscopic investigation, with the ability to in-situmanipulating the individual atoms or molecules. It can also work jointly with various near-field or external field characterization techniques, making it a most important technique for precisely detecting and manipulating quantum properties at a single molecule level. In this paper, we review recent research progress of quantum states of surface-supported single molecules and relevant structures based on scanning tunneling microscopy. We start from the methods for the synthesis of molecular structures with desired quantum states, and then we review the recent advances in the local spin states for single molecular systems and the optical properties of single molecules serving as a single-photon source. An emerging family of molecular nanographene systems showing intriguing topological properties and magnetic properties is also reviewed. In the last part, we summarize the research progress made recently and prospect the future development of the quantum states at a single molecular level.