You can shoot starlings and the English sparrow anywhere in the USA all year long - no season or limits. These are both non-native species to the States.
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There is a crow season in Washington State, check your game laws for exact dates. They can be shot all year around when found in the act of depredation. There is no bag limit during the crow hunting season. In the spring I shoot them if they are after bird nests.Crows ARE NOT a protected species. Ravens are. I shoot crows all the time on a couple of dairy farms where I pest. I also shoot some 'game' birds but that is on the farm and while they are eating a pooping on the food piles. On the farm...except for rapters robins swallows, etc., it it flies it dies.