Error vector magnitude (EVM) has proven to be one of the optical performance monitoring metrics providing the quantitative estimation of error statistics. However, the EVM estimation efficiency has not been fully exploited in terms of complexity and energy consumption. Therefore, in this paper, we explore two deep-learning-based EVM estimation schemes. The first scheme exploits convolutional neural networks (CNNs) to extract EVM information from images of the constellation diagram in the in-phase/quadrature (IQ) complex plane or amplitude histograms (AHs). The second scheme relies on feedforward neural networks (FFNNs) extracting features from a vectorized representation of AHs. In both cases, we use short sequences of 32 Gbaud m-ary quadrature amplitude modulation (mQAM) signals captured before or after a carrier phase recovery. The impacts of the sequence length, neural network structure, and data set representation on the EVM estimation accuracy as well as the model training time are thoroughly studied. Furthermore, we validate the performance of the proposed schemes using the experimental implementation of 28 Gbaud 64QAM signals. We achieve a mean absolute estimation error below 0.15%, with short signals consisting of only 100 symbols per IQ cluster. Considering the estimation accuracy, the implementation complexity, and the potential energy savings, the proposed CNN- and FFNN-based schemes can be used to perform time-sensitive and accurate EVM estimation for mQAM signal quality monitoring purposes.