Fortunately... swords have been more or less forgotten by most state laws. I don't know of any restrictions on ownership. Carrying might be problematic in some places, and nomatter what the law says... will generate lots of unwanted attention. :-)
I have several swords, spears, daggers and other medieval era weapons that I display at home. Some people are surprised to find out that yes, they're all perfectly legal. Funny what people think about the law, sometimes...
A number of places have banned knives over a certain blade length, often 6”, 5 1/2”, 4”, I have even heard of a ban down to an unbelievable 1 1/2 inches!