A woman in Washington State did just that and was shocked to find her Social Security Number and her name listed on a web site. Even worse was the site: it was owned and operated by the State of Washington's Secretary of State. In fact, the same site lists names and Social Security Number of other state residents as well.
KATU Television then went looking for more and they found them. At last count, more than 42,000 people had their Social Security numbers online -- about half of them from Spokane, Wash.
You can read more on the KATU web site at http://www.katu.com/news/specialreports/95041939.html
NOTE: I did find my Social Security Number online, but it was listed in an ad that contained hundreds of other random Social Security Numbers, all without any identifying information. That web site sells records obtained from credit reports and is obviously trying to "con" you into paying for information that is available to you from other sources free of charge. In fact, none of the listed Social Security Numbers include any identifying information. The site is simply placing random numbers online and hoping that someone will "bite" for their paid services.
I wouldn't worry about non-identifying SSN numbers placed randomly on a site. If you find your name included with the number, that is another issue!