@goldenstateairguner and @gendoc Yes it’s confusing. After @MarkT posted a reference number for the statute that explains the exceptions I looked it up, read, and posted information that I located as well as the source in this thread. Within the link is a document dated October 2021 from the United States Department of the Interior (Fish and Wildlife) with information that defines and discusses the exceptions to the Migratory Bird Treaties Act. Beyond that I think it best to speak with your local game wardens (and/or local and state officials that enforce your hunting laws) and use your own discretion based upon your understanding of the information that is available to the public. 
 
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In my area as a kid there was a bounty on crows. 25 cents, would buy you 2 cola's and a candy bar. A few years ago hunters were encouraged to shoot Cowbirds. They lay eggs in song birds nest and their chick kicks out the song birds eggs. The was a drop in song bird then. Now song bird numbers are better and Cowbirds are protected again. Now on the pest side. Raccoons have a bad reputation and if the get on your property you can pretty much kill them no question asked.. But if you trap them and move them to a place that would bother anything you can go to jail for up to 3 years. That is just crazy save an animal and you go to jail. You can't not kill or trap a squirrel out of season unless you obtain an additional permit and you have to be a business. The list goes on and most of the time the rule was written by people that never had problems with pest and most rules certainly don't help the average person. But still if we have a hunting license we need to follow the rules and work on changing them, I am sorry to say.


 
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Looks like in MN there are three crow seasons. 

crow2.1650767893.jpg


Also, this is from the MN DRN regulations, and is interesting both because it seems to open the doors wide open, and also because it explicitly enumerates the weapons that can be used to take a crow:

crow1.1650767157.jpg


If I am reading this correctly, airguns can only be used on unprotected birds and mammals.

unprotected.jp.1650768067.JPG

 
Out of curiosity does anyone know why most state even have a crow season! I never heard of anyone eat a crow so why a season? Seems to me a pest permit would do for most.

@qball Do I know? No. But I’ve thought about this for a while and what stands out most to me is agriculture. In regions that produce food crops, crows have potential access to a steady supply of food. I suspect this is especially applicable where monoculture of a fruit, vegetable, or nut crops are common (think corn, peanuts, beans, peas, etc). They don’t just eat the aforementioned crops. If you’ve ever planted a garden then you may know that they will scratch up your seeds, some types of seedlings, and eat them. 


If there’s a suitable habitat and plenty of access to food and water with little effective defense it’s highly plausible that their numbers can get out of hand rather quickly. Hence the need for control in order to prevent large numbers of corvids from decimating food crops among other things (like eating feed for livestock). Crows are opportunistic in my observation. They seem to want an easy meal. In my area I’ve seen farmers try different methods to keep them out of family food plots that don’t involve killing. There is one method I’ve seen that is somewhat effective at deterring crows from damaging and eating crops in small plots, but it doesn’t seem feasible for commercial parcels. This is my opinion based upon observation and limited knowledge and experience. And I have heard of people eating them. Strange thing, isn’t it?

Edited to add reference and source. 
For reference: 

764709FB-4FB9-4C1D-90FE-A09F1EABE24B.1650984888.jpeg


Source of above screen shot - https://www.canr.msu.edu/outreach/wildlife-crows.pdf
 
The question is does any state that has a season or no closed season allow them to be taken with a air gun? Seems most limit method of take to the same restrictions as all other migratory game birds. Strange because that is a federal law. But the feds turn around and say in the case of crows, blk birds, grackels etc. No limitations of any kind on methods of take. Most states ignore that. And air gunners are locked out from inclusion.
 
I live in NY and the only 2 birds that can legally be shot are Starlings and HOSPS, everything else is protected. While I can't legally shoot them, there's nothing that says I can't fire a warning shot to chase them off and it works. They're actually very smart even if ugly pests. Scare the bejesus out of the scout and he warns the rest to stay away.
Not true. Many birds can be shot in NY state. The below link gives some examples and seasons...

 
In Ohio, season is closed from April something to June 1st. I got one a few weeks ago with an air rifle. I carry two calls with me when walking the dogs. A crow call and a rabbit squealer. Crows come in to both. Not so hard hunting them with a shotgun but using an airgun adds another dimension. You have to hide and take long shots..

They aren't any dumber down here in West Central Ohio man, let me tell ya.
If they see you, game over.

Complete concealment and zero movement are the rule of the day when calling in crows.