A distributed algorithm is self-stabilizing if after faults and attacks hit the system and place it in some arbitrary global state, the system recovers from this catastrophic situation without external intervention in finite time. In this paper, we consider the problem of constructing self-stabilizingly a locally maximizable task (such as constructing a maximal independent set, a maximal matching, or a grundy coloring) in uniform unidirectional networks of arbitrary shape. On the negative side, we present evidence that in uniform networks, deterministic self-stabilization of this problem is impossible. Also, the silence property (i.e. having communication fixed from some point in every execution) is impossible to guarantee, either for deterministic or for probabilistic variants of protocols. %X On the positive side, we present a series of generic protocols that can be instantiated for all considered locally maximizable tasks. First, we design a deterministic protocol for arbitrary unidirectional networks with unique identifiers that exhibits polynomial space and time complexity in asynchronous scheduling. We complement the study with probabilistic protocols for the uniform case: the first probabilistic protocol requires infinite memory but copes with asynchronous scheduling, while the second probabilistic protocol has polynomial space complexity but can only handle synchronous scheduling. Both probabilistic solutions have expected polynomial time complexity.