ICPSR Search
Searches one of the largest dataset repositories in the world. Be sure to filter to public availability to make sure you get data you can download.
Data.gov
Most datasets come from a government organization of some kind. This searches through those that are open to public use.
U.S. Census demographics
Starting point for any kind of people demographic: economic, income, poverty, housing, race, population, age, education, etc. Can be adjusted for national, state, or local areas.
Bureau of Labor Statistics
Employment, unemployment, wages, prices, consumer spending, etc.
National Archive of Criminal Justice Data (NACJD)
Searches the repository of a multitude of criminal justice datasets. After a search, be sure to limit results to "public use" to get those you can freely download.
Agency for Healthcare Research and Quality (AHRQ data)
Health, health costs, access to health, quality of care, insurance, etc.
Bureau of Justice Statistics
Major repository of criminal justice data. Crime, corrections, recidivism, victims of crime, sentencing, etc.
Texas Academic Performance Reports
Information on the performance of students in each school and district in Texas every year. Data is at the state, region, district, or campus level. Inlcudes students demographics, ethnicity, socioeconomic status, performance on state tests, etc. Choose year then look for TAPR Download link.
Correlates of State Policy
Over 3000 variables at the state level for all 50 states going back 10 or more years. Excellent source for researching state differences or evaluating policy by investigating how one variable relates to another.
IPPSR
Michigan State's repository of public policy datasets. Arranged by general subjects.