Fourier-inspired single-pixel holography (FISH) is an effective digital holography (DH) approach that utilizes a single-pixel detector instead of a conventional camera to capture light field information. FISH combines the Fourier single-pixel imaging and off-axis holography technique, allowing one to acquire useful information directly, rather than recording the hologram in the spatial domain and filtering unwanted terms in the Fourier domain. Furthermore, we employ a deep learning technique to jointly optimize the sampling mask and the imaging enhancement model, to achieve high-quality results at a low sampling ratio. Both simulations and experimental results demonstrate the effectiveness of FISH in single-pixel phase imaging. FISH combines the strengths of single-pixel imaging (SPI) and DH, potentially expanding DH's applications to specialized spectral bands and low-light environments while equipping SPI with capabilities for phase detection and coherent gating.