Virtually all of them, and most by federal and state law.
Almost all birds native to the United States, whether migratory or not, are protected under the Migratory Bird Treaty Act. This includes songbirds, hummingbirds, woodpeckers, roadrunners, birds of prey, waterfowl, wading birds, shorebirds, seabirds, etc.
State laws can be more restrictive than federal but not less, so the MBTA represents the minimum level of protection for the native birds it covers—and it’s one of the strongest wildlife laws in the world. It was enacted in response to the wholesale slaughter of egrets, herons, and other charismatic birds for the feather trade. (It also put an end to market hunting of native birds, but this came too late to help the Passenger Pigeon.)
The MBTA allows the U.S. Fish & Wildlife Service to issue permits for managed recreational hunting of most traditional prey species, lethal control of “nuisance” birds, and live capture and possession of raptors for falconry (in states where falconry is legal), but commercial exploitation is strictly prohibited. That means you can buy a state hunting license and federal and state “duck stamps” and legally shoot your limit of waterfowl, but you can’t sell them or any part of thereof (including mounted trophies). With some exceptions for migratory gamebirds (including captive-bred ornamental waterfowl and the aforementioned taxidermy), you also can’t keep protected birds in captivity (even for altruistic reasons) or have their feathers, bones, nests, or eggs in your possession without a scientific, educational, or religious/cultural permit.
Native “upland/nonmigratory gamebirds” —quail, grouse, Wild Turkey, and (in Texas only) Plain Chachalaca—are under state jurisdiction. States often regulate hunting of introduced game species such as pheasants and partridges, but most nonnative species—House Sparrows, European Starlings, Rock (Domestic/Feral) Pigeons, Eurasian Collared-Doves, Indian (Blue) Peafowl, feral chickens, feral parrots, escaped (but non-breeding) domestic* or pet birds, etc.—have little or no legal protection anywhere in the U.S. except whatever might be afforded by county or city ordinances (which may have priority over state ordinances).
Disclaimer: IANAL. This is just a brief summary of common situations where the MBTA applies. It’s up to you to read federal, state, and local laws and understand how they apply to your situation. If in doubt, consult the appropriate government agency for advice.