We present the first part of our Disks ARound T Tauri Stars with SPHERE (DARTTS-S) survey: observations of eight T Tauri stars that were selected based on their strong (sub)millimeter excesses using SPHERE/IRDIS polarimetric differential imaging in the J and H bands. All observations successfully detect the disks, which appear vastly different in size, from approximate to 80 au in scattered light to >400 au, and display total polarized disk fluxes between 0.06% and 0.89% of the stellar flux. For five of these disks, we are able to determine the three-dimensional structure and the flaring of the disk surface, which appears to be relatively consistent across the different disks, with flaring exponents a between approximate to 1.1 and approxim...