This paper aims to collect, analyse and compare the quality data regarding the current status and prospective evolution of search engine multi- sided platforms offered by leading platform companies. We describe the search multi-sided platforms competition across different markets in a general way and compare several emerging markets where Google multi-sided platform is not a leader. The methodology and organization of search platform ontology/taxonomy, multi-sided platform business model pattern, and search platform business model pattern are discussed. Then detailed case analysis of search multi-sided platform business models is developed for Yandex search multi-sided platform. Finally, the possible ways of further evolution of global and local search multi-sided platforms is discussed.