The learning and usage of an API is supported by official documentation. Like source code, API documentation is itself a software product. Several research results show that bad design in API documentation can make the reuse of API features difficult. Indeed, similar to code smells or code antipatterns, poorly designed API documentation can also exhibit 'smells'. Such documentation smells can be described as bad documentation styles that do not necessarily produce an incorrect documentation but nevertheless make the documentation difficult to properly understand and to use. Recent research on API documentation has focused on finding content inaccuracies in API documentation and to complement API documentation with external resources (e.g., crowd-shared code examples). We are aware of no research that focused on the automatic detection of API documentation smells. This paper makes two contributions. First, we produce a catalog of five API documentation smells by consulting literature on API documentation presentation problems. We create a benchmark dataset of 1,000 API documentation units by exhaustively and manually validating the presence of the five smells in Java official API reference and instruction documentation. Second, we conduct a survey of 21 professional software developers to validate the catalog. The developers agreed that they frequently encounter all five smells in API official documentation and 95.2% of them reported that the presence of the documentation smells negatively affects their productivity. The participants wished for tool support to automatically detect and fix the smells in API official documentation. We develop a suite of rule-based, deep and shallow machine learning classifiers to automatically detect the smells. The best performing classifier BERT, a deep learning model, achieves F1-scores of 0.75 - 0.97.