The Transformer model has achieved state-of-the-art performance in many sequence modeling tasks. However, how to leverage model capacity with large or variable depths is still an open challenge. We present a probabilistic framework to automatically learn which layer(s) to use by learning the posterior distributions of layer selection. As an extension of this framework, we propose a novel method to train one shared Transformer network for multilingual machine translation with different layer selection posteriors for each language pair. The proposed method alleviates the vanishing gradient issue and enables stable training of deep Transformers (e.g. 100 layers). We evaluate on WMT English-German machine translation and masked language modeling tasks, where our method outperforms existing approaches for training deeper Transformers. Experiments on multilingual machine translation demonstrate that this approach can effectively leverage increased model capacity and bring universal improvement for both many-to-one and one-to-many translation with diverse language pairs.