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Phase singularities are loci of darkness surrounded by monochromatic light in a scalar field, with applications in optical trapping, super-resolution imaging, and structured light-matter interactions. Although 1D singular structures, such as optical vortices, are the most common due to their robust topological properties, uncommon 0D (point) and 2D (sheet) singular structures can be generated by wavefront-shaping devices such as metasurfaces. Here, using the design flexibility of metasurfaces, we deterministically position ten identical point singularities in a cylindrically symmetric field generated by a single illumination source. The phasefront is inverse-designed using phase gradient maximization with an automatically-differentiable propagator. This process produces tight longitudinal intensity confinement. The singularity array is experimentally realized with a 1 mm diameter TiO2 metasurface. One possible application is blue-detuned neutral atom trap arrays, for which this light field would enforce 3D confinement and a potential depth around 0.22 mK per watt of incident trapping laser power. Metasurface-enabled point singularity engineering may significantly simplify and miniaturize the optical architecture required to produce super-resolution microscopes and dark traps.