Image enhancement methods are applied to medical images to reduce the noise that they contain. There are many academic studies in the literature using classical image enhancement methods. Ultrasound imaging is a medical imaging method that is used for the diagnosis of diseases. In this study, speckle noises with Rayleigh distribution at four different noise levels (σ = 0.10, 0.25, 0.50, 0.75) are added to ultrasound images of the brachial plexus nerve region. Five different deep learning networks (Dilated Convolution Autoencoder Denoising Network/Di-Conv-AE-Net, Denoising U-Shaped Net/D-U-Net, BatchRenormalization U-Net/Br-U-Net, Generative Adversarial Denoising Network/DGan-Net, and CNN Residual Network/DeRNet) are used for reducing the speckle noises of the ultrasound images. The performances of the deep networks are compared with block-matching and 3D filtering (BM3D), which is one of the most preferred classical image enhancement algorithms; with classical filters including Bilateral, Frost, Kuan, Lee, Mean, and Median Filters; and with deep learning networks including Learning Pixel-Distribution Prior with Wider Convolution for Image Denoising (WIN5-RB), Denoising Prior Driven Deep Neural Network for Image Restoration (DPDNN), and Fingerprint Image Denoising and Inpainting Using M-Net Based Convolutional Neural Networks (FPD-M-Net). Network performance is evaluated according to peak signal-to-noise ratio (PSNR), structural similarity index (SSIM), and runtime criteria and the proposed deep learning networks are shown to outperform the other networks.