Compression of digital holograms allows one to store, transmit, and reconstruct large sets of holographic data. There are many digital image compression methods, and usually wavelets are used for this task. However, many significant specialties exist for compression of digital holograms. As a result, it is preferential to use a set of methods that includes filtering, scalar and vector quantization, wavelet processing, etc. These methods in conjunction allow one to achieve an acceptable quality of reconstructed images and significant compression ratios. In this paper, wavelet compression of amplitude/phase and real/imaginary parts of the Fourier spectrum of filtered off-axis digital holograms is compared. The combination of frequency filtering, compression of the obtained spectral components, and extra compression of the wavelet decomposition coefficients by threshold processing and quantization is analyzed. Computer-generated and experimentally recorded digital holograms are compressed. The quality of the obtained reconstructed images is estimated. The results demonstrate the possibility of compression ratios of 380 using real/imaginary parts. Amplitude/phase compression allows ratios that are a factor of 2–4 lower for obtaining similar quality of reconstructed objects.