Solr is widely used in many different scenarios: from the well-known, big, new sites such as The Guardian (http://www.guardian.co.uk/) to an application as popular as Instagram (http://instagr.am/). It has also been adopted by big companies such as Apple, Disney, or Goldman Sachs; and there have been some very specific adoptions such as the search over http://citeseerx.ist.psu.edu/network for scientific citations.
Two very widely adopted use cases are aggregators and metasearch engines and Open Catalog Access (OPAC). The first type requires continuous indexing over sparse data and makes a business out of being able to capture users by its indexes, and the second generally needs a read-only exposition of metadata with powerful search. Good examples of online catalogs that have used Solr for a long time are Internet Archive (http://www.archive.org/), the well-known open digital library, and VuFind (http://www.vufind.org/), an open source discovery portal for libraries.
Other very common use cases include news sites, institutional sites such as USA government sites, a publisher site, and others. Even if every site will not necessarily require full-text and the other features of Solr, the use cases can fill a very long list.
For a more extended and yet incomplete list of projects and sites that are using Solr in the world, please refer to the following page in the official documentation: