Singular value decomposition (SVD)-based ultrasound blood flow clutter filters have recently demonstrated substantial improvement in clutter rejection for ultrafast plane wave microvessel imaging, and have become the commonly used clutter filtering method for many novel ultrafast imaging applications such as functional ultrasound and super-resolution imaging. At present, however, the computational burden of SVD remains as a major hurdle for practical implementation and clinical translation of this method. To address this challenge, in the study we present two blood flow clutter filtering methods based on randomized SVD (rSVD) and randomized spatial downsampling to accelerate SVD clutter filtering with minimal compromise to the clutter filter performance. rSVD accelerates SVD computation by approximating the k largest singular values, while random downsampling accelerates both full SVD and rSVD by decomposing the original large data matrix into small matrices that can be processed in parallel. An in vitro blood flow phantom study with the presence of heavy tissue clutter showed significantly improved computational performance using the proposed methods with minimal deterioration to the clutter filter performance (less than 3-dB reduction in blood to clutter ratio, less than 0.2-cm(2)/s(2) increase in flow mean squared error, less than 0.1-cm/s increase in the standard deviation of the vessel blood flow signal, and less than 0.3-cm/s increase in tissue clutter velocity for both full SVD and rSVD when the downsampling factor was less than 20x). The maximum acceleration was about threefold from randomized spatial downsampling, and approximately another threefold from rSVD. An in vivo rabbit kidney perfusion study showed that rSVD provided comparable performance to full SVD in clutter rejection in vivo (maximum difference of blood to clutter ratio was less than 0.6 dB), and random downsampling provided artifact-free perfusion imaging results when combined with both full SVD and rSVD. The blood to clutter ratio was still above 10 dB with a downsampling factor of 60x. We also demonstrated realtime microvessel imaging feasibility (similar to 40-ms processing time) by combining rSVD with random downsampling. The findings and methods presented in this paper may greatly facilitate the new area of ultrafast microvessel imaging research.