We perform a detailed investigation of sources from the Cold Cores Catalogue of Planck Objects (C3PO). Our goal is to probe the reliability of the detections, validate the separation between warm and cold dust emission components, provide the first glimpse at the nature, internal morphology and physical characterictics of the Planck-detected sources. We focus on a sub-sample of ten sources from the C3PO list, selected to sample different environments, from high latitude cirrus to nearby (150pc) and remote (2kpc) molecular complexes. We present Planck surface brightness maps and derive the dust temperature, emissivity spectral index, and column densities of the fields. With the help of higher resolution Herschel and AKARI continuum observations and molecular line data, we investigate the morphology of the sources and the properties of the substructures at scales below the Planck beam size. The cold clumps detected by Planck are found to be located on large-scale filamentary (or cometary) structures that extend up to 20pc in the remote sources. The thickness of these filaments ranges between 0.3 and 3pc, for column densities NH2 ~ 0.1 to 1.6 × 1022 cm-2, and with linear mass density covering a broad range, between 15 and 400 M&sun; pc-1. The dust temperatures are low (between 10 and 15K) and the Planck cold clumps correspond to local minima of the line-of-sight averaged dust temperature in these fields. These low temperatures are confirmed when AKARI and Herschel data are added to the spectral energy distributions. Herschel data reveal a wealth of substructure within the Planck cold clumps. In all cases (except two sources harbouring young stellar objects), the substructures are found to be colder, with temperatures as low as 7K. Molecular line observations provide gas column densities which are consistent with those inferred from the dust. The linewidths are all supra-thermal, providing large virial linear mass densities in the range 10 to 300 M&sun; pc-1, comparable within factors of a few, to the gas linear mass densities. The analysis of this small set of cold clumps already probes a broad variety of structures in the C3PO sample, probably associated with different evolutionary stages, from cold and starless clumps, to young protostellar objects still embedded in their cold surrounding cloud. Because of the all-sky coverage and its sensitivity, Planck is able to detect and locate the coldest spots in massive elongated structures that may be the long-searched for progenitors of stellar clusters. Appendix A is available in electronic form at http://www.aanda.orgCorresponding author: I. Ristorcelli, e-mail: isabelle.ristorcelli [at] irap.omp.eu (isabelle[dot]ristorcelli[at]irap[dot]omp[dot]eu)