The estimation of visual complexity in 3D remains a relatively under-studied problem. An aspect of this deficiency is due to the lack of suitable publicly available benchmarks. In this work, three datasets are used to investigate this problem from different angles: (1) a subset of the ABC dataset of CAD shapes, augmented with perceived visual complexity rankings; (2) an existing primate tooth dataset, with complexity rankings derived from established domain attributes; and (3) a novel dataset of fractal shapes with perceived visual complexity rankings. Ground truth rankings for (1) and (3) were derived from human judgments obtained from two studies in which lay individuals performed two-alternative forced-choice comparisons between pairs of shapes. We then invited researchers to submit results for algorithms designed to model shape complexity. We find that most methods that operate directly in 3D yield low, positive correlations; whereas-despite working on 2D projections derived from the original geometry-the best results were obtained by a subset of 2D methods.