Many universities/research institutions have a Dataverse instance (its open-source software). Find them here.

"This is a list of repositories and databases for open data." Part of the Open Access Directory.

"OpenDOAR is the quality-assured, global Directory of Open Access Repositories. You can search and browse through thousands of registered repositories based on a range of features, such as location, software or type of material held."

"A Comprehensive List of Open Data Portals from Around the World"

"QDR selects, ingests, curates, archives, manages, durably preserves, and provides access to digital data used in qualitative and multi-method social inquiry."

"re3data.org is a global registry of research data repositories that covers research data repositories from different academic disciplines. It presents repositories for the permanent storage and access of data sets to researchers, funding bodies, publishers and scholarly institutions."

This organization conducts polls on politics, the media and American perspectives. They also conduct the Pew Global Attitudes Project, "a series of worldwide public opinion surveys that encompasses a broad array of subjects ranging from people's assessments of their own lives to their views about the current state of the world and important issues of the day. More than 90,000 interviews in 50 countries have been conducted as part of the project's work"

US state and city open data portals. Not comprehensive!

"... a resource center for gender equality data, with statistics, tools and reference materials covering various topics."
Locate other World Bank data portals at its Data Portals and Tools site.

"PUMS USA collects, preserves and harmonizes U.S. census microdata and provides easy access to this data with enhanced documentation. Data includes decennial censuses from 1790 to 2010 and American Community Surveys (ACS) from 2000 to the present."

"Why does America vote as it does on Election Day? The mission of the American National Election Studies (ANES) is to inform explanations of election outcomes by providing data that support rich hypothesis testing, maximize methodological excellence, measure many variables, and promote comparisons across people, contexts, and time. The ANES serves this mission by providing researchers with a view of the political world through the eyes of ordinary citizens."
"To serve the research needs of social scientists, teachers, students, policy makers and journalists, the ANES produces high quality data from its own surveys on voting, public opinion, and political participation."

"The U.S. Religion Census is your source for religious data at the county level. It reports the number of congregations in every U.S. county and equivalents for participating faith groups."

We believe that a key reason why we fail to achieve the progress we are capable of is that we do not make enough use of this existing research and data: the important knowledge is often stored in inaccessible databases, locked away behind paywalls and buried under jargon in academic papers.
The goal of our work is to make the knowledge on the big problems accessible and understandable. As we say on our homepage, Our World in Data is about Research and data to make progress against the world’s largest problems."

"...a continuously-updated database of thousands of citations of works using data held in the ICPSR archive. The works include journal articles, books, book chapters, government and agency reports, working papers, dissertations, conference papers, meeting presentations, unpublished manuscripts, magazine and newspaper articles, and audiovisual materials."

"The European Social Survey (ESS) is an academically driven cross-national survey that has been conducted across Europe since its establishment in 2001. Every two years, face-to-face interviews are conducted with newly selected, cross-sectional samples."

"This is a bibliography of publications that use data from the Roper Center. Some cite information on overall responses obtained from iPOLL; some are based on individual-level analyses of datasets held by the Roper Center, and others use our Presidential approval series or other indexes constructed from Roper Center data." Research articles only.
Note: We no longer have access to the Roper Center or its iPoll database, but this bibliography is still useful.

"Designed to guide users to the content they need to learn a little or a lot about their method, SAGE Research Methods includes more than 1000 books, reference works, journal articles, and instructional videos by world-leading academics from across the social sciences, including the largest collection of qualitative methods books available online from any scholarly publisher. The Methods Map can help those less familiar with research methods to find the best technique to use in their research."