Single shot SPIFI operating in the Fourier Domain is demonstrated for the first time to our knowledge. We present initial results capturing Fourier Domain single shot images with both one and two-dimensional detectors and demonstrate that the same enhanced frequency support that is characteristic of classical SPIFI translates directly into single shot SPIFI architectures as well. Linear Fourier Domain single shot SPIFI is systematically analyzed, for both types of detectors. Finally, we show that despite the complex pulse structure imposed on the illumination beam by SPIFI, nonlinear single shot SPIFI can be realized, and third harmonic generation imaging is demonstrated. Prepared by LLNL under Contract DE-AC52-07NA27344. LLNL-ABS-860152.