Context. Cyclopropenone was first detected in the cold and less dense envelope of the giant molecular cloud Sagittarius B2(N). It was found later in several cold dark clouds and it may be possible to detect its minor isotopic species in these environments. In addition, the main species may well be identified in warmer environments.Aims. We aim to extend existing line lists of isotopologs of c-H2C3O from the microwave to the millimeter region and create one for the singly deuterated isotopolog to facilitate their detections in space. Furthermore, we aim to extend the line list of the main isotopic species to the submillimeter region and to evaluate an equilibrium structure of the molecule.Methods. We employed a cyclopropenone sample in natural isotopic composition to investigate the rotational spectra of the main and O-18-containing isotopologs as well as the two isotopomers containing one C-13 atom. Spectral recordings of the singly and doubly deuterated isotopic species were obtained using a cyclopropenone sample highly enriched in deuterium. We recorded rotational transitions in the 70-126 and 160-245 GHz regions for all isotopologs and also in the 342-505 GHz range for the main species. Quantum-chemical calculations were carried out to evaluate initial spectroscopic parameters and the differences between ground-state and equilibrium rotational parameters in order to derive semi-empirical equilibrium structural parameters.Results. We determined new or improved spectroscopic parameters for six isotopologs and structural parameters according to different structure models.Conclusions. The spectroscopic parameters are accurate enough to identify minor isotopic species at centimeter and millimeter wavelengths while those of the main species are deemed to be reliable up to 1 THz. Our structural parameters differ from earlier ones. The deviations are attributed to misassignments in the earlier spectrum of one isotopic species.