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Multiple scenario cooperative games model situations where the worth of the coalitions is simultaneously valued in several scenarios or under different states of nature. For multiple scenario cost allocation games, we focus on the identification of those allocations which are stable in the sense that agents have no incentives to abandon the group. The stability of an allocation depends on how the quantities assigned to the coalitions are compared with the vector-valued worth of the coalition. Therefore, different extensions of the notion of core emerge, depending on how these comparisons are made. In this paper different notions of core for multiple scenario cost allocation games are studied, and the relationships between them and the existing core solutions are analyzed. We also address the inclusion in the model of partial information about the relative importance of the scenarios which often refers to the probabilities of occurrence. In order to identify allocations which are also stable in the presence of partial information, we extend the notions of core to this new setting, and provide results which permit the calculation of the corresponding sets of allocations. The potential applicability of our analysis is illustrated with the allocation of the total cost of the waste collection service between the different urban areas in the city of Seville. [ABSTRACT FROM AUTHOR]