We have determined an approach for simultaneous reconstruction and denoising of 3D seismic data with randomly missing traces. The core in simultaneous reconstruction and denoising of 3D seismic data is the choice of constraint method. Recently, there have been two types of popular approaches to choose such a constraint: sparsity-promoting transforms using a sparsity constraint and rank reduction methods using a rank constraint. Although the sparsity-promoting transform enjoys the direct advantage of high efficiency, it lacks adaptivity to a variety of data patterns. On the other hand, the rank reduction method can be adaptively applied to different data sets, but its computational cost is quite high. We investigate multiple constraints for simultaneous seismic data reconstruction and denoising based on a novel hybrid rank-sparsity constraint (HRSC) model, which aims at combining the benefits of the sparsity-promoting transforms and rank reduction methods. Also, we design the corresponding HRSC algorithmic framework to effectively solve our new model via tightly combining a sparsity-promoting transform and a rank reduction method, which is more powerful in simultaneous reconstruction and denoising of 3D seismic data. Our HRSC framework aims at providing an extra level of constraint and, thus, can significantly improve the signal-to-noise ratio (S/N) of the reconstructed results with higher efficiency. Application of the HRSC framework on synthetic and field 3D seismic data demonstrates superior performance in terms of S/N and visual observation compared with the well-known rank reduction method, known as multichannel singular spectrum analysis.