"[A] survey has shown that over 95% of scientists in evolution and ecology think that data should be publicly archived..." Whitlock MC, McPeek MA, Rausher MD, Rieseberg L, Moore AJ. Data archiving. 2010. American Naturalist 175:145-146. DOI: 10.1086/650340 Unfortunately, thought has not been turned to reality yet. It is remarkably difficult to find useful data sets. Here are few sites and tips you may find useful. If you discover others, please share.
Tip: When accessing online journal articles, examine the table of contents of the journal for links to "supplementary materials." Sometimes you will miss those links if you go directly to the PDF versions.
New PubMed search option "hasdatabanklist" can help identify articles with associated data files. More information. Most of the records with this tag are for Genebank and other NIH/NLM/NCBI databanks, but figshare and Dryad are also included in the Medline databank sources. If you want to avoid genetics-related files, try [your subject/keywords] AND hasdatabanklist AND (dryad OR figshare).