This paper studies the conditions for existence and stability of stationaryclusterstructures in lattices of diffusivelycoupled dynamical systems within the framework of a new interpretation of cluster synchronization as classical synchronization of clusteroscillators (C-oscillators). The study of existence of cluster attractors is based on the linear chains of clusteroscillators, defining possible types of clusterstructures in chains. First, we present interval estimates for the range of coupling strengths in which cluster attractors can exist. Then we formulate and prove the basic theorems about the local stability of the various clusterstructures. The presented methodology can be extended to study clusterstructures on lattices of different geometry and forms such as linear clusterstructures in two-dimensional lattices, layered clusterstructures in three-dimensional lattices and clusterstructures in ring-shaped systems.