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 approximate to 1.6. We also confirm literature results with regard to the inclination and position angle of several of our disks and are able to determine which side is the near side of the disk in most cases. While there is a clear trend of disk mass with stellar ages (approximate to 1 to >10Myr), no correlations of disk structures with age were found. There are also no correlations with either stellar mass or submillimeter flux. We do not detect significant differences between the J and H bands. However, we note that while a high fraction (7/8) of the disks in our sample show ring-shaped substructures, none of them display spirals, in contrast to the disks around more massive Herbig Ae/Be stars, where spiral features are common.