Beyond these useful examples from specific projects, there are an increasing number of repositories for existing social network data, which may also be useful for considering what your own aims may be. Here is a list of a few of those:
Lin Freeman’s collection of “classic” SN datasets.
Katya Ognyanova’s collection of collections.
The Colorado Index of Complex Networks (ICON).
Inter University Consortium for Political and Social Research; ICPSR is not network specific, but is a good general resource, and does include quite a few network datasets.
The Stanford Large Network (of mostly Online) Dataset Collection.