The fractal dimension and multifractal spectrum can characterize the complexity of fracture sets. However, studies of impacts of fracture geometries on their fractal and multifractal characteristics are largely insufficient, especially for three-dimensional (3-D) fracture networks (natural fractures are always 3-D instead of 2-D). In this work, we construct 3-D stochastic discrete fracture networks with an open-source DFN software, HatchFrac. Systematical investigations are then conducted to study the impact of geometrical fracture properties and system sizes on the fractal and multifractal characteristics. The box-counting method is adopted to calculate the fractal dimension and multi-fractal descriptors. The fractal dimension, D, and the difference of the singularity exponent, ∆α, represent the fractal and multifractal patterns, respectively. Two critical (percolative and over-percolative) stages of fracture networks are considered. 3-D fracture networks share similar characteristics with 2-D fracture networks at percolation. However, results at an over-percolative stage are systematically different. At the first stage, fracture orientations (κ), lengths (a) and system sizes (L) have positive correlations with D and ∆α. D is weakly correlated with fracture positions (FD), meaning that the fractal dimension is insensitive to clustering effects. However, ∆α is strongly correlated with FD, implying that ∆α can characterize the heterogeneity caused by clustering effects. a and L are positively correlated with ∆α, and κ and FD have negative correlations. At stage two, the sensitivity results on D are similar to stage one, but a and L become negatively correlated with ∆α. Impacts of κ and FD become more significant.