In many natural language processing (NLP) tasks the same input (e.g. source sentence) can have multiple possible outputs (e.g. translations). To analyze how this ambiguity (also known as intrinsic uncertainty) shapes the distribution learned by neural sequence models we measure sentence-level uncertainty by computing the degree of overlap between references in multi-reference test sets from two different NLP tasks: machine translation (MT) and grammatical error correction (GEC). At both the sentence- and the task-level, intrinsic uncertainty has major implications for various aspects of search such as the inductive biases in beam search and the complexity of exact search. In particular, we show that well-known pathologies such as a high number of beam search errors, the inadequacy of the mode, and the drop in system performance with large beam sizes apply to tasks with high level of ambiguity such as MT but not to less uncertain tasks such as GEC. Furthermore, we propose a novel exact n-best search algorithm for neural sequence models, and show that intrinsic uncertainty affects model uncertainty as the model tends to overly spread out the probability mass for uncertain tasks and sentences.