  Near-IR polarimetric images of protoplanetary disks enable us to characterize
substructures that might be due to the interaction with (forming) planets. The
available census is strongly biased toward massive disks around old stars,
however. The DARTTS program aims at alleviating this bias by imaging a large
number of T Tauri stars with diverse properties. In this work, we present new
SPHERE images of 21 circumstellar disks, which is the largest sample released
to date. The targets of this work are significantly younger than those
published thus far with polarimetric near-IR (NIR) imaging. Scattered light is
unambiguously resolved in 11 targets, and some polarized unresolved signal is
detected in 3 additional sources. Some disk substructures are detected.
However, the paucity of spirals and shadows from this sample reinforces the
trend according to which these NIR features are associated with Herbig stars,
either because they are older or more massive. Furthermore, disk rings that are
apparent in ALMA observations of some targets do not appear to have
corresponding detections with SPHERE. Inner cavities larger than 15 au are also
absent from our images, even though they are expected from the spectral energy
distribution. On the other hand, 3 objects show extended filaments at larger
scale that are indicative of strong interaction with the surrounding medium.
All but one of the undetected disks are best explained by their limited size
(less than 20 au), and the high occurrence of stellar companions in these
sources suggests an important role in limiting the disk size. One undetected
disk is massive and very large at millimeter wavelengths, implying that it is
self-shadowed in the NIR. This work paves the way toward a more complete and
less biased sample of scattered-light observations, which is required to
interpret how disk features evolve throughout the disk lifetime.
