Context. Some luminous and ultraluminous infrared galaxies (LIRGs and ULIRGs) host extremely compact (r < 100 pc) and dusty nuclei. The high extinction associated with large column densities of gas and dust toward these objects render them hard to detect at many wavelengths. The intense infrared radiation arising from warm dust in these sources can provide a significant fraction of the bolometric luminosity of the galaxy and is prone to excite vibrational levels of molecules such as HCN. This results in emission from the rotational transitions of vibrationally excited HCN (HCN-vib); the brightest emission is found in compact obscured nuclei (CONs; ΣHCN − vib > 1 L⊙ pc−2 in the J = 3−2 transition). However, there have been no systematic searches for CONs, and it is unknown how common they are. Aims. We aim to establish how common CONs are in the local Universe (z < 0.08), and whether their prevalence depends on the luminosity or other properties of the host galaxy. Methods. We conducted an Atacama Large Millimeter/submillimeter Array survey of the rotational J = 3−2 transition of HCN-vib in a volume-limited sample of 46 far-infrared luminous galaxies. Results. Compact obscured nuclei are identified in 38−13+18% of the ULIRGs, 21−6+12% of the LIRGs, and 0−0+9% of the lower luminosity galaxies. We find no dependence on the inclination of the host galaxy, but strong evidence of lower IRAS 25 μm to 60 μm flux density ratios (f25/f60) in CONs (with the exception of one galaxy, NGC 4418) compared to the rest of the sample. Furthermore, we find that CONs have stronger silicate features (s9.7 μm), but similar polycyclic aromatic hydrocarbon equivalent widths (EQW6.2 μm) compared to other galaxies. Along with signatures of molecular inflows seen in the far-infrared in most CONs, submillimeter observations also reveal compact, often collimated, outflows. Conclusions. In the local Universe, CONs are primarily found in (U)LIRGs, in which they are remarkably common. As such systems are often highly disturbed, inclinations are difficult to estimate, and high-resolution continuum observations of the individual nuclei are required to determine if the CON phenomenon is related to the inclinations of the nuclear disks. Further studies of the in- and outflow properties of CONs should also be conducted to investigate how these are connected to each other and to the CON phenomenon. The lower f25/f60 ratios in CONs as well as the results for the mid-infrared diagnostics investigated (EQW6.2 μm and s9.7 μm) are consistent with the notion that large dust columns gradually shift the radiation from the hot nucleus to longer wavelengths, making the mid- and far-infrared “photospheres” significantly cooler than the interior regions. Finally, to assess the importance of CONs in the context of galaxy evolution, it is necessary to extend this study to higher redshifts where (U)LIRGs are more common.