- A lattice of traps for ultracold neutral atoms is a promising tool for experimental investigation in quantum physics and quantum information processing. We consider regular arrays of thin film type-II superconducting nanodisks, with only one pinned vortex in each of them, and also arrays of mesoscopic disks, each containing many vortices whose distribution is characterized by the superconducting current density. In both cases we show theoretically that the induced magnetic field can create a 3D lattice of magnetic traps for cold atoms without any additional bias field. Applying a bias DC field parallel to the superconductor surface, one can control the depth and sizes of the traps, their heights above the chip surface, potential barriers between the traps, as well as the structure and dimension of the lattices. In the adiabatic approximation the atom cloud shape is represented by the shape of a closed iso-surface of the magnetic field magnitude chosen in accordance with the atom cloud temperature. The computed trap sizes, heights and the distances between the neighboring traps are typically from tens to hundreds nanometers for nanodisks and of the order of 1 μm for mesoscopic disks. Our calculations show that the depth of magnetic traps on mesoscopic disks is, typically, between 0.3 G and 7.6 G; for the nanodisks the depth is about 0.3 G.