In a systematic study, we compare the density statistics in high-resolution numerical experiments of supersonic isothermal turbulence, driven by the usually adopted solenoidal (divergence-free) forcing and by compressive (curl-free) forcing. We find that for the same rms Mach number, compressive forcing produces much stronger density enhancements and larger voids compared to solenoidal forcing. Consequently, the Fourier spectra of density fluctuations are significantly steeper. This result is confirmed using the Delta-variance analysis, which yields power-law exponents beta similar to 3.4 for compressive forcing and beta similar to 2.8 for solenoidal forcing. We obtain fractal dimension estimates from the density spectra and Delta-variance scaling, and by using the box counting, mass size, and perimeter area methods applied to the volumetric data, projections, and slices of our turbulent density fields. Our results suggest that compressive forcing yields fractal dimensions significantly smaller compared to solenoidal forcing. However, the actual values depend sensitively on the adopted method, with the most reliable estimates based on the Delta-variance, or equivalently, on Fourier spectra. Using these methods, we obtain D similar to 2.3 for compressive and D similar to 2.6 for solenoidal forcing, which is within the range of fractal dimension estimates inferred from observations (D similar to 2.0-2.7). The velocity dispersion to size relations for both solenoidal and compressive forcings obtained from velocity spectra follow a power law with exponents in the range 0.4-0.5, in good agreement with previous studies.