Warning, lawyer pedantry, no legal advice offered or implied:
The "You'll Shoot Your Eye Out" brigade is right that there is a statute in Delaware about this, title 11, section 1445 of the Delaware Code. "Unlawfully dealing with a dangerous weapon" includes "A person is guilty of unlawfully dealing with a dangerous weapon when: (1) The person possesses, sells or in any manner has control of: (a). A weapon which by compressed air or by spring discharges or projects a pellet, slug or bullet, except a BB or air gun which does not discharge or project a pellet or slug larger than a BB shot; or (b). A pellet, slug or bullet, intending that it be used in any weapon prohibited by paragraph (1)a. of this section..." and also a few other things. Possession of a banned "dangerous weapon" is an unclassified misdemeanor, but giving an air rifle to an unsupervised child can be a felony.
The statute has been on the books in basically the same form since the early 1970s. I have no idea if people ever get prosecuted for possessing illegal air rifles, because that's not my practice area.
Pedantic point #1: What does "larger than a BB shot" mean? Does it mean "of larger caliber than BB shot"? That definitely bans .22 caliber air rifles, and possibly .177 air rifles too, because round BB shot is a couple thousandths of an inch less in diameter than .177 air rifle pellets (even though .177 air rifle pellets are commonly sold labeled as "BB").
Does it mean "heavier than BB shot"? No standard weight for BB shot, depends on what the shot's made of. The ranges of weights for BB shot and for .177 air rifle pellets definitely overlap. The ranges for BB shot and .22 cal pellets might as well, I don't know.
Does it mean "longer along the axis of travel than BB shot"? That's pretty much every commercially available air rifle banned right there.
Or does "a BB shot" mean just "ammunition for what normal people call a BB gun, including .22 air rifles"? Beats me.
Pedantic point 2: Airsoft. Most Airsoft ammo is larger in caliber than .177, but lighter in weight than most BB shot. Note also, the statute only goes to a "weapon which [fires] by compressed air or by spring." Arguably not electric Airsoft guns (which I don't believe existed in the 1970s).
End pedantry, commence rant.
One more dumb trap for the unwary of dubious constitutionality and negligible sanity. BB guns are OK, and any adult who can pass a background check can buy a rimfire or centerfire firearm that can do a lot more damage if misused, but .22 cal air rifles are so evil and also so ill-suited for 2nd Amendment purposes that they have to be and can be banned?