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We present novel planar permanent magnet designs that create points above the plane where the magnitude of the static magnetic field is a local nonzero magnetic field point minimum. The designs represent potential magnetic resonance microscopy ''lenses'' where only a point region of the sample located in the ''focus'' spot of the structures is resonant. The high magnetic field curvatures of the ''lenses'' ensure that the regions outside of the ''focus'' spot are inactive or filtered out in the resonance detection process. In contrast to the original magnetic resonance ''lens'' design that required an applied uniform external bias field, the new designs are self-biased. Each planar permanent magnet design is accompanied by the equivalent planar single noncrossing conductor design. We also introduce the concept of Amperean current doubling in planar perpendicularly magnetized thin films that can be used to improve the performance of each permanent magnet design we present. We experimentally constructed macroscale representative permanent magnet and wire structures and measured their magnetic field properties that confirm our numerical predictions. Single conductor current-carrying designs are suitable for single layer lithographic fabrication, as we also experimentally demonstrate. Finally, we present the case that nanometer scale recording of perpendicular anisotropy thin magnetic films using presently available data storage technology can potentially provide the ultimate miniaturization of the presented designs.