We have proposed a novel fast projection onto convex sets (FPOCS) solver for compressive sensing of seismic data via sparsity constraint in seislet transform domain. The seislet transform is demonstrated to be much sparser than other state-of-the-art sparse transforms and thus is more suitable for a compressive sensing based seismic data recovery approach. The FPOCS can obtain much faster convergence than conventional POCS, which can potentially make the seislet-based POCS approach applicable in practice according to the efficiency acceleration. We have found that the the POCS based approach can be superior than the IST based approach in relatively cleaner dataset while can be slightly worse than the IST based approach in relatively noisier dataset. This conclusion can guide us to use different iterative approach according to the noise level in the data. In addition to the signal-to-noise ratio (SNR), the local similarity is also used to measure the data recovery performance. We are surprisingly to find out that even in the case of very close SNRs, the local similarity can still show slight but obvious difference, and thus the local similarity measurement is more sensitive in comparing different state-of-the-art approaches. The seislet transform based compressive sensing can achieve obviously better data recovery results than $f-k$ transform based scenarios because of a much sparser structure in the seislet transform domain. We have used both synthetic and field data examples to demonstrate the superior performance of the proposed seislet-based FPOCS approach.