In Ontario (Canada) you can hunt Crows year round.
If you are hunting during a open gun season for big game (Deer, Moose, Bear) you are pretty much restricted to hunting with a shotgun (with #2 or smaller lead or BBB or smaller steel shot) UNLESS you also have a big game licence (just want to stop guys from trying to shoot a deer when they have no licence I guess ??).
Otherwise you can use whatever you want.
Some counties have a maximum caliber size of .275 (or smaller), so the largest practical CF you could use would be a 270 Win/Wby or WSM etc. That restriction has been in place since "before memory" and nobody can really answer why the caliber limit.
Crows fall under our "small game" regulations but there is no limit on them as there are with rabbits, partridge etc. They are mostly considered pests that facilitate the spread of Lime disease and in the Province of Manitoba some of the local governments actually have "full time staff" doing nothing but hunting crows.