This work presents a planar antenna with a rectangular shape designed over a Moore curve fractal-shaped High Impedance Surface (HIS). The Moore fractal geometries are space-filling curves and are useful for multiband applications. The Moore curve-shaped fractal HIS is simulated up to three iterations, and performance is examined. The proposed antenna has multiband operation within the S-band, C-band, and lower X-band frequency of operation. The antenna has a peak gain of 5.08 dB, 4.69 dB, and 5.07 dB with a Moore curve fractal HIS, with iterations 1, 2, and 3 used as the ground plane. The antenna has been analyzed regarding the reflection coefficient, radiation pattern, 3-D polar plots, and surface current distribution. With Moore curve iteration 1, a shaped HIS provides a maximum bandwidth of 740 MHz with the center frequency of 10.95 GHz, 1.24 GHz with the center frequency of 10.57 GHz, and 1.09 GHz with the center frequency of 12.5 GHz with the second and third iterations, respectively.