I'm no expert on American law, but I doubt you would be able to just walk about with a sword within a scabbard on your belt/back.
Carrying it wrapped up in a bag or something having just bought it, sure. But I expect you would get a few funny looks.
As usual, that depends on where you are.
Chances are, the bluer or more urban your state is the more likely it is somebody thought of that and outlawed it. Interestingly, in the Corporate Oligarchy of Delaware (my old home state) an unconcealed sword was perfectly legal as it was unconcealed. State law for "deadly weapons" included wordage about concealing them (naughty) and using them on people (very naughty) but not carring them around in broad daylight. Which I routinely did, given the circumstance (not a six foot long katana or scottish claymore, of course, more of a machetelike implement).
I never got hassled for it. Of course, I also once got beat up by a cop for having a walking stick, which he swore he thought was a shotgun. Which carrying unconcealed is also legal.
So, your mileage will vary.
Insofar as using it in your home it's not a case of anything goes but certainly a case of more goes than out on the street. I can't picture a very long sword being good for much but thrusting (nothing like putting a big gash in your drywall before you even hit the guy) but somehow I'm imagining the excretion of bricks of a perp jimmies my door to find a lean, muscular, angry kung-fu looking dude with no shirt, a very nice katana, and a Surefire in his teeth.