This page includes links to datasets and documentation that I have used in a variety of statistics classes. The list is a work in progress.
There are also a wide variety of data sources freely available online.
Health Sciences
National Center for Health Statistics (Center for Disease Control) - links to publications and data organized by disease or health outcome
Critical care database (requires completing a short human subjects training course and application for approval)
Scott Emerson's collection of (relatively old) clinical datasets and documentation
Social Sciences
United Nations data (includes tourism, industry, education, environment, crime, population demographics, and more)
Social sciences data from the Inter-university Consortium for Political and Social Research (ICPSR) - the thematic collections can give you some ideas about topics and starting points.
College Scorecard data
data.gov: U.S. government open data https://www.data.gov/
Other Sciences
Sports
Baseball: The Guide to Sabermetric research from the Society for American Baseball Research recommends several data sources and explains what is available from each
Basketball: Kaggle NCAA data (requires a free account), data.world basketball datasets
Variety Databases
Europa Open Data Portal -- data ranging from a grassland butterfly index, to housing construction prices, to social beliefs
Google's Public Data Explorer to locate and visualize publicly available data
UCI machine learning repository: https://archive.ics.uci.edu/ml/datasets.php
Kaggle: https://www.kaggle.com/
FiveThirtyEight (yes, it's a data visualization blog, but they also make much of their data public!): https://data.fivethirtyeight.com/
GitHub: Awesome Public Datasets https://github.com/awesomedata/awesome-public-datasets
Datasets from Buzzfeed News on Github https://github.com/BuzzFeedNews
All datasets available in R packages are listed at https://vincentarelbundock.github.io/Rdatasets/datasets.html.