This paper presents two novel approaches to speckle reduction in SAR images. The former relies on the multiplicative speckle model as an MMSE filtering performed in the wavelet domain by means of an adaptive shrinkage of the detail coefficients of an undecimated decomposition. Each coefficient is shrunk by the variance ratio of the noise-free coefficient to the noisy one. All the above quantities are analytically calculated from the speckled image, the noise variance, and the wavelet filters only, without resorting to any model to describe the underlying backscatter. Estimation of the local statistics driving the filter is expedited and layered processing allows to extend adaptivity also across the spatial scale. The latter is not model-based and provides a blind estimation of the backscatter underlying the speckled image stated as a problem of matching pursuits. The local adaptive MMSE estimator is obtained as an expansion in series of a finite number of prototype estimators, fitting the spatial features of the different statistical classes encountered, e.g., edges and textures. Such estimators are calculated in a fuzzy fashion through an automatic training procedure. The space-varying coefficients of the expansion are stated as degrees of fuzzy membership of a pixel to each of the estimators. A thorough performance comparison is carried out with the Gamma-MAP filter and with the Rational Laplacian Pyramid (RLP) filter, recently introduced by three of the authors. On simulated speckled images both the proposed filters gain almost 3 dB SNR with respect to conventional local-statistics (Lee/Kuan) filtering. Experiments carried out on widespread test SAR images and on a speckled mosaic image, comprising synthetic shapes, textures, and details from true SAR images, demonstrate that the visual quality of the results is excellent in terms of both background smoothing and preservation of edge sharpness, textures, and point targets. The absence of decimation in the wavelet decomposition avoids the typical ringing impairments produced by critically-sampled wavelet-based denoising.