I am trying to search for papers such that the data used in that paper are also published. This is because sometimes I really enjoy replicating a paper and it is somehow hard to ask the author to give you the data especially if he surveyed this data. I mainly use it to see if I may able to replicate and get better in my coding skills. It is also better to know if the data are available upon request.
The academic climate is more and more open, more and more publishers are requiring data to be deposit and openly available alongside the papers once they are published. A good example is PLOSOne, "data availability" is listed on every paper, detailing where and how to obtain the data if its openly available, and reason for withholding if it's not.
Another way is to reverse your process and look for the open data sets, since every open data published usually have a study (sometimes more than one) relate to it (cited by the paper). An example, the high-energy physics open data platform HEPData, any and all data sets published here are linked by DOI to a publication. These data might be more complicated to process, if you are interested to explore, a lot of HEP data tools are provided on CERN Open Data. A similar data publishing platform for earth and environmental science data is PANGEA
You can also look for data sets of your particular interest on RE3DATA, a registry for research data repositories. They might not all be openly accessible, but many are.
Outside of the scientific data arena, many governmental departments publish open data on their website, pay special attention to the census and statistics pages. But so far I haven't found a search engine that's dedicated for open government data, which is a shame.
Have fun exploring open data!