Much research effort has been devoted to explaining the success of deep learning. Random Matrix Theory (RMT) provides an emerging way to this end: spectral analysis of large random matrices involved in a trained deep neural network (DNN) such as weight matrices or Hessian matrices with respect to the stochastic gradient descent algorithm. To have more comprehensive understanding of weight matrices spectra, we conduct extensive experiments on weight matrices in different modules, e.g., layers, networks and data sets. Following the previous work of \cite{martin2018implicit}, we classify the spectra in the terminal stage into three main types: Light Tail (LT), Bulk Transition period (BT) and Heavy Tail(HT). These different types, especially HT, implicitly indicate some regularization in the DNNs. A main contribution from the paper is that we identify the difficulty of the classification problem as a driving factor for the appearance of heavy tail in weight matrices spectra. Higher the classification difficulty, higher the chance for HT to appear. Moreover, the classification difficulty can be affected by the signal-to-noise ratio of the dataset, or by the complexity of the classification problem (complex features, large number of classes) as well. Leveraging on this finding, we further propose a spectral criterion to detect the appearance of heavy tails and use it to early stop the training process without testing data. Such early stopped DNNs have the merit of avoiding overfitting and unnecessary extra training while preserving a much comparable generalization ability. These findings from the paper are validated in several NNs, using Gaussian synthetic data and real data sets (MNIST and CIFAR10).