In the last few years, Twitter has become a powerful tool for publishing and discussing information. Yet, content exploration in Twitter requires substantial effort. Users often have to scan information streams by hand. In this paper, we approach this problem by means of faceted search. We propose strategies for inferring facets and facet values on Twitter by enriching the semantics of individual Twitter messages (tweets) and present different methods, including personalized and context-adaptive methods, for making faceted search on Twitter more effective. We conduct a large-scale evaluation of faceted search strategies, show significant improvements over keyword search and reveal significant benefits of those strategies that (i) further enrich the semantics of tweets by exploiting links posted in tweets, and that (ii) support users in selecting facet value pairs by adapting the faceted search interface to the specific needs and preferences of a user.