Neural decoders are being explored for 5G and beyond communication networks which have ultra-low-latency and energy-efficiency requirements. We consider several ultra-short linear and non-linear block codes and design neural decoders for them, which (i) provide near maximum-likelihood (ML) performance in terms of bit and block error rates, and (ii) have low decoding complexity (and hence, latency and energy consumption) in terms of the number of hidden layers and hidden nodes. We explore (i) single-label, (ii) multi-label, and (iii) concatenated coding based decoders. We find that single-label decoders with a single hidden layer having a sufficient number of nodes achieve near-ML performance. Since the number of output nodes required in single-label decoders is exponential in the message length, we investigate a multi-label neural decoder requiring a significantly lower number of output nodes but having decreased performance in terms of block error rate. To eliminate extra errors, we concatenate an outer code with the original code as the inner code. The inner and outer codes are decoded by multi-label and single-label neural decoders, respectively, leading to better block error performance than that is achievable using multi-label decoders.