In my experience, if the "sword" is not sharpened and does not have a metal edge that can be dangerous, it is not likely illegal. However, the final decision is left up to the police, prosecutor and ultimately it is up to a Judge and/or jury to decide. I believe you're safe having it on your wall.
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Under New York Penal Law Sec. 265.01 (Criminal possession of a weapon in the fourth degree)
A person is guilty of criminal possession of a weapon in the fourth
(1) He possesses any firearm, electronic dart gun, electronic stun
gun, gravity knife, switchblade knife, pilum ballistic knife, metal
knuckle knife, cane sword, billy, blackjack, bludgeon, metal knuckles,
chuka stick, sand bag, sandclub, wrist-brace type slingshot or
slungshot, shirken or "Kung Fu star"...
It is a Class A Misdemeanor punishable by no more than one year in prison. I doubt a broken wooden sword in your home would ever be written up as a violation of the penal law.