We cannot do this using the Facebook website itself
But Facebook API does let you do that How to do it is explained here:
https://developers.facebook.com/docs/reference/api/search/
One of the websites that has implemented this is: www.openstatussearch.com
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There are 3 ways of doing this:
Quoting from the link above
We support search for the following types of objects:
All public posts: https://graph.facebook.com/search?q=watermelon&type=post This search type does not support limit/offset paging.
Now develop your own parser for this data. That's one way of doing this.
Using JavaScript SDK https://developers.facebook.com/docs/javascript/gettingstarted/ With the fb.dataquery method. What you want to do is use the stream table to get the status posts for users.
Using PHP SDK (I personally prefer this) https:/developers.facebook.com/docs/php/gettingstarted/ Decode (using json_decode) the json object recevied by this url: https://graph.facebook.com/search?q=Vacation+in+italy&type=post
Update: Version 2.0 of the GraphAPI does not let you search for posts anymore. See the updated documentation.
You can continue to search for posts by changing your API call to:
https://graph.facebook.com/v1.0/search?q=search_query&type=post&access_token={access_token}