This paper investigates the use of multi-distribution deep neural networks (MD-DNNs) for automatic lexical stress detection and pitch accent detection, which are useful for suprasegmental mispronunciation detection and diagnosis in second-language (L2) English speech. The features used in this paper cover syllable-based prosodic features (including maximum syllable loudness, syllable nucleus duration and a pair of dynamic pitch values) as well as lexical and syntactic features (encoded as binary variables). As stressed/accented syllables are more prominent than their neighbors, the two preceding and two following syllables are also taken into consideration. Experimental results show that the MD-DNN for lexical stress detection achieves an accuracy of 87.9% in syllable classification (primary/secondary/no stress) for words with three or more syllables. This performance is much better than those of our previous work using Gaussian mixture models (GMMs) and the prominence model (PM), whose accuracies are 72.1% and 76.3% respectively. Approached similarly as the lexical stress detector, the pitch accent detector obtains an accuracy of 90.2%, which is better than the results of using the GMMs and PM by about 9.6% and 6.9% respectively.