Context. The physical state of cold cloud clumps has a great impact on the process and efficiency of star formation and the masses of the forming stars inside these objects. The sub-millimetre survey of the Planck space observatory and the far-infrared follow-up mapping of the Herschel space telescope provide an unbiased, large sample of these cold objects. Aims. We have observed 12 CO(1−0) and 13 CO(1−0) emission in 35 high-density clumps in 26 Herschel fields sampling different environments in the Galaxy. We derive the physical properties of the objects and estimate their gravitational stability. Methods. The densities and temperatures of the clumps were calculated from both the dust continuum and the molecular line data. Kinematic distances were derived using 13 CO(1−0) line velocities to verify previous distance estimates and the sizes and masses of the objects were calculated by fitting 2D Gaussian functions to their optical depth distribution maps on 250 µm. The masses and virial masses were estimated assuming an upper and lower limit on the kinetic temperatures and considering uncertainties due to distance limitations. Results. The derived excitation temperatures are between 8.5−19.5 K, and for most clumps between 10−15 K, while the Herschel-derived dust colour temperatures are more uniform, between 12−16 K. The sizes (0.1−3 pc), 13 CO column densities (0.5−44 × 10 15 cm −2 ) and masses (from less than 0.1 M to more than 1500 M ) of the objects all span broad ranges. We provide new kinematic distance estimates, identify gravitationally bound or unbound structures and discuss their nature. Conclusions. The sample contains objects on a wide scale of temperatures, densities and sizes. Eleven gravitationally unbound clumps were found, many of them smaller than 0.3 pc, but large, parsec-scale clouds with a few hundred solar masses appear as well. Colder clumps have generally high column densities but warmer objects appear at both low and higher column densities. The clump column densities derived from the line and dust observations correlate well, but are heavily affected by uncertainties of the dust properties, varying molecular abundances and optical depth effects.