Medical image semantic segmentation is essential in computer-aided diagnosis systems. It can separate tissues and lesions in the image and provide valuable information to radiologists and doctors. The breast ultrasound (BUS) images have advantages: no radiation, low cost, portable, etc. However, there are two unfavorable characteristics: (1) the dataset size is often small due to the difficulty in obtaining the ground truths, and (2) BUS images are usually in poor quality. Trustworthy BUS image segmentation is urgent in breast cancer computer-aided diagnosis systems, especially for fully understanding the BUS images and segmenting the breast anatomy, which supports breast cancer risk assessment. The main challenge for this task is uncertainty in both pixels and channels of the BUS images. In this paper, we propose a Spatial and Channel-wise Fuzzy Uncertainty Reduction Network (SCFURNet) for BUS image semantic segmentation. The proposed architecture can reduce the uncertainty in the original segmentation frameworks. We apply the proposed method to four datasets: (1) a five-category BUS image dataset with 325 images, and (2) three BUS image datasets with only tumor category (1830 images in total). The proposed approach compares state-of-the-art methods such as U-Net with VGG-16, ResNet-50/ResNet-101, Deeplab, FCN-8s, PSPNet, U-Net with information extension, attention U-Net, and U-Net with the self-attention mechanism. It achieves 2.03%, 1.84%, and 2.88% improvements in the Jaccard index on three public BUS datasets, and 6.72% improvement in the tumor category and 4.32% improvement in the overall performance on the five-category dataset compared with that of the original U-shape network with ResNet-101 since it can handle the uncertainty effectively and efficiently.