  The structure formation of the dense interstellar material and the
fragmentation of clumps into cores is a fundamental step to understand how
stars and stellar clusters form. We aim to establish a statistical view of
clump fragmentation at sub-parsec scales based on a large sample of massive
clumps selected from the ATLASGAL survey. We used the APEX/SABOCA camera at 350
$\mu$m to image clumps at a resolution of 8.$''$5. The majority of the sample
consists of massive clumps that are weak or in absorption at 24 $\mu$m. We
resolve rich filamentary structures and identify the population of compact
sources. We use association with mid-infrared 22-24 $\mu$m and 70 $\mu$m point
sources to pin down the star formation activity of the cores. We then
statistically assess their physical properties, and the fragmentation
characteristics of massive clumps. We find a moderate correlation between the
clump fragmentation levels with the clump gas density and the predicted number
of fragments with pure Jeans fragmentation scenario; we find a strong
correlation between the mass of the most massive fragment and the total clump
mass, suggesting that the self-gravity may play an important role in the
clumps' small scale structure formation. We identify 27 massive quiescent cores
with $M_{\rm core}>100$ M$_{\odot}$ within 5 kpc; these are massive enough to
be self-gravitating but do not yet show any sign of star-formation. This sample
comprises, therefore, promising candidates of massive pre-stellar cores, or
deeply embedded high-mass protostars.

Astrophysics of Galaxies;Solar and Stellar Astrophysics