Context. The structure formation of the dense interstellar material and the fragmentation of clumps into cores is a fundamental step for understanding how stars and stellar clusters form. Aims. We aim to establish a statistical view of clump fragmentation at subparsec scales based on a large sample of massive clumps selected from the ATLASGAL survey. Methods. We used the APEX/SABOCA camera at 350 μm to image clumps at a resolution of 8_.^″5, corresponding to physical scales of < 0.2 pc at a distance < 5 kpc. The majority of the sample consists of massive clumps that are weak or in absorption at 24 μm. We resolved spherical and filamentary structures and identified the population of compact sources. Complemented with archival Herschel data, we derived the physical properties, such as dust temperature, mass, and bolometric luminosity of clumps and cores. We used association with mid-infrared 22−24 μm and 70 μm point sources to determine the star formation activity of the cores. We then statistically assessed their physical properties and the fragmentation characteristics of massive clumps. Results. We detect emission at 350 μm toward all targets and find that it typically exhibits a filamentary (-like) morphology and hosts a population of compact sources. Using Gaussclumps, we identify 1120 compact sources and derive the physical parameters and star formation activity for 971 of these, 874 of which are associated with 444 clumps. We find a moderate correlation between the clump fragmentation levels with the clump gas density and the predicted number of fragments with a pure Jeans fragmentation scenario. We find a strong correlation between the mass of the most massive fragment and the total clump mass, suggesting that self-gravity may play an important role in the small-scale structure formation of the clumps. Finally, due to the improved angular resolution compared to ATLASGAL, we are able to identify 27 massive quiescent cores with M_core > 100 M_⊙ within 5 kpc; these are massive enough to be self-gravitating, but do not yet show any sign of star formation. This sample therefore comprises promising candidates of massive prestellar cores or deeply embedded high-mass protostars. Conclusions. The submillimeter observations of the massive clumps that are weak or completely dark at 24 μm reveal rich filamentary structures and an embedded population of compact cores. The maximum core mass is likely determined by the self-gravity of the clump. The rarity of massive prestellar core candidates implies short collapse timescales for dense structures.