Singular value decomposition of ultrafast imaging ultrasonic data sets has recently been shown to build a vector basis far more adapted to the discrimination of tissue and blood flow than the classical Fourier basis, improving by large factor clutter filtering and blood flow estimation. However, the question of optimally estimating the boundary between the tissue subspace and the blood flow subspace remained unanswered. Here, we introduce an efficient estimator for automatic thresholding of subspaces and compare it to an exhaustive list of thirteen estimators that could achieve this task based on the main characteristics of the singular components, namely the singular values, the temporal singular vectors, and the spatial singular vectors. The performance of those fourteen estimators was tested in vitro in a large set of controlled experimental conditions with different tissue motion and flow speeds on a phantom. The estimator based on the degree of resemblance of spatial singular vectors outperformed all others. Apart from solving the thresholding problem, the additional benefit with this estimator was its denoising capabilities, strongly increasing the contrast to noise ratio and lowering the noise floor by at least 5 dB. This confirms that, contrary to conventional clutter filtering techniques that are almost exclusively based on temporal characteristics, efficient clutter filtering of ultrafast Doppler imaging cannot overlook space. Finally, this estimator was applied in vivo on various organs (human brain, kidney, carotid, and thyroid) and showed efficient clutter filtering and noise suppression, improving largely the dynamic range of the obtained ultrafast power Doppler images.