In recent years, many studies have examined filters for eliminating or reducing speckle noise, which is inherent to ultrasound images, in order to improve the metrological evaluation of their biomedical applications. In the case of medical ultrasound images, said noise can produce uncertainty in the diagnosis because details, such as limits and edges, should be preserved. Most algorithms can eliminate speckle noise, but they do not consider the conservation of these details. This paper describes, in detail, 27 techniques that mainly focus on the smoothing or elimination of speckle noise in medical ultrasound images. The aim of this study is to highlight the importance of improving said smoothing and elimination, which are directly related to several processes (such as the detection of regions of interest) described in other articles examined in this study. Furthermore, the description of this collection of techniques facilitates the implementation of evaluations and research with a more specific scope. This study initially covers several classical methods, such as spatial filtering, diffusion filtering, and wavelet filtering. Subsequently, it describes recent techniques in the field of machine learning focused on deep learning, which are not yet well known but greatly relevant, along with some modern and hybrid models in the field of speckle-noise filtering. Finally, five Full-Reference (FR) distortion metrics, common in filter evaluation processes, are detailed along with a compensation methodology between FR and Non-Reference (NR) metrics, which can generate greater certainty in the classification of the filters by considering the information of their behavior in terms of perceptual quality provided by NR metrics.