In recent years data-driven machine learning approaches have been extensively studied to replace or enhance traditionally model-based processing in digital communication systems. In this work, we focus on equalization and propose a novel neural network (NN-)based approach, referred to as SICNN. SICNN is designed by deep unfolding a model-based iterative soft interference cancellation (SIC) method. It eliminates the main disadvantages of its model-based counterpart, which suffers from high computational complexity and performance degradation due to required approximations. We present different variants of SICNN. SICNNv1 is specifically tailored to single carrier frequency domain equalization (SC-FDE) systems, the communication system mainly regarded in this work. SICNNv2 is more universal and is applicable as an equalizer in any communication system with a block-based data transmission scheme. Moreover, for both SICNNv1 and SICNNv2, we present versions with highly reduced numbers of learnable parameters. Another contribution of this work is a novel approach for generating training datasets for NN-based equalizers, which significantly improves their performance at high signal-to-noise ratios. We compare the bit error ratio performance of the proposed NN-based equalizers with state-of-the-art model-based and NN-based approaches, highlighting the superiority of SICNNv1 over all other methods for SC-FDE. Exemplarily, to emphasize its universality, SICNNv2 is additionally applied to a unique word orthogonal frequency division multiplexing (UW-OFDM) system, where it achieves state-of-the-art performance. Furthermore, we present a thorough complexity analysis of the proposed NN-based equalization approaches, and we investigate the influence of the training set size on the performance of NN-based equalizers.