We report on an algorithm for fast wavefront sensing that incorporates sparse representation for the first time in practice. The partial derivatives of optical wavefronts were sampled sparsely with a Shack-Hartman wavefront sensor (SHWFS) by randomly subsampling the original SHWFS data to as little as 5%. Reconstruction was performed by a sparse representation algorithm that utilized the Zernike basis. We name this method sparse Zernike (SPARZER). Experiments on real and simulated data attest to the accuracy of the proposed techniques as compared to traditional sampling and reconstruction methods. We have made the corresponding dataset and software freely available online. Compressed wavefront sensing offers the potential to increase the speed of wavefront acquisition and to defray the cost of SHWFS devices.