We offer predictions of symmetron modified gravity in the neighborhood of realistic dark matter halos. The predictions for the fifth force (FF) are obtained by solving the nonlinear symmetron equation of motion in the spherical NFW approximation. We compare the three major known screening mechanisms: Vainshtein, Chameleon, and Symmetron around such dark matter sources, emphasizing the significant differences between them and highlighting observational tests which exploit these differences. In addition to halos, we investigate the behavior of the FF in voids in chameleon modified gravity models using the spherical collapse method. The FF can be many times larger than the Newtonian force. This is very different from the case in halos, where the FF is no more than 1/3 of gravity. Individual voids in chameleon models grow larger by ~10%. The number density is up to 2.5 times larger in chameleon models. This difference is about 10 times larger than that in the halo mass function. Turning to weak lensing data analysis, we search for the lensing signal of massive filaments between 220,000 pairs of Luminous Red Galaxies (LRGs) from the Sloan Digital Sky Survey. We use a nulling technique to remove the contribution of the LRG halos, resulting in a 10-sigma detection of the filament lensing signal. We compare the measurements with halo model predictions based on a calculation of 3-point halo-halo-mass correlations. Comparing the "thick" halo model filament to a "thin" string of halos, thick filaments larger than a Mpc in width are clearly preferred by the data. In addition to filaments, dark matter voids should exhibit a weak lensing signal. We find voids in the galaxy distribution using a novel algorithm, then perform a stacked shear measurement on 20,000 voids with radii between 15-40 Mpc/h and redshifts between 0.16-0.37. We detect the characteristic radial shear signal of voids with a statistical significance that exceeds 13-sigma. The mass profile corresponds to a fractional underdensity of about -0.4 inside the void radius and a slow approach to the mean density.