Advances in deep learning hold promise for predicting clinical factors from human brain images. In this study, we applied a spherical harmonics-based convolutional neural network approach (SPHARM-Net) to MRI-derived brain shape metrics to predict age, sex, and Alzheimer's disease (AD) diagnosis. MRI-derived brain features included vertex-wise cortical curvature, convexity, thickness, and surface area. SPHARM-Net performs convolutions using the spherical harmonic transforms, eliminating the need to explicitly define neighborhood size, and achieving rotational equivariance. Sex classification and age regression were carried out in a large sample of healthy adults (UK Biobank; N=32,979), and AD classification performance was tested in a large, publicly available sample (ADNI; N=1,213). SPHARM-Net showed strong performance for sex classification (accuracy=0.91; balanced accuracy= 0.91; AUC=0.97), and age regression (average absolute error=2.97 years; R-squared=0.77; Pearson's coefficient=0.9). AD classification also performed well (accuracy=0.86; balanced accuracy=0.83; AUC=0.9). Our experiments demonstrate promising preliminary performance using the SPHARM-Net for two widely studied benchmarking tasks and for AD classification. Future work will include comparisons of shape-based methods and extending these analysis to more challenging tasks such as mood disorder classification.