Government agencies are by far the largest keepers of statistics on social and economic subjects. Often, locating a particular statistic is first a matter of figuring out what agency has jurisdiction over that subject area, then combing their web site for relevant data.
Local agencies may gather statistics that they don't publish on their web sites, so a phone call to the agency might turn up data otherwise unavailable.
Private organizations (often non-profits) also keep statistics that are relevant to their mission. KIDS COUNT is an example of an organization that collates government statistics about children and also gathers its own statistics. If you know of an advocacy organization for the subject area you're investigating, be sure to check their Web site for data.
Keep in mind that some figures are notoriously difficult to produce. Figures about the numbers of homeless, for example, or undocumented immigrants, are at best only estimates, because many people in these groups don't wish to be counted. Certain types of crime, such as human trafficking, are similarly difficult to track because the activities are so secret.
If Googling a statistic, try including some of these keywords in your search: