Many self-supervised speech models (S3Ms) have been introduced over the last few years, producing performance and data efficiency improvements for a variety of speech tasks. Evidence is emerging that different S3Ms encode linguistic information in different layers, and also that some S3Ms appear to learn phone-like sub-word units. However, the extent to which these models capture larger linguistic units, such as words, and where word-related information is encoded, remains unclear. In this study, we conduct several analyses of word segment representations extracted from different layers of three S3Ms: wav2vec2, HuBERT, and WavLM. We employ canonical correlation analysis (CCA), a lightweight analysis tool, to measure the similarity between these representations and word-level linguistic properties. We find that the maximal word-level linguistic content tends to be found in intermediate model layers, while some lower-level information like pronunciation is also retained in higher layers of HuBERT and WavLM. Syntactic and semantic word attributes have similar layer-wise behavior. We also find that, for all of the models tested, word identity information is concentrated near the center of each word segment. We then test the layer-wise performance of the same models, when used directly with no additional learned parameters, on several tasks: acoustic word discrimination, word segmentation, and semantic sentence similarity. We find similar layer-wise trends in performance, and furthermore, find that when using the best-performing layer of HuBERT or WavLM, it is possible to achieve performance on word segmentation and sentence similarity that rivals more complex existing approaches.