Some solid posts in this thread already, especially killgar's. I wanted to get into something else too, which is if you actually use a knife in self-defense.
An important distinction is that we are talking about several different types of laws. The laws most have referred to so far are about carry and intent of carry, both statutory and practical (cop's opinion). These cover situations where you have not actually used the knife, but are questioned or searched by law enforcement.
But if you're talking to the cops because you used the knife in some way in relation to another human being, we're talking another kettle of fish entirely. States do not have clear, hardline laws or rules about self-defense in the way you seem to be thinking (i.e. merely a matter of the weapon used). Rather, self-defense is an affirmative defense you raise in court when you are accused by the state of having committed a crime of brandishing, aggravated assault, or murder. And your ability to succeed at mounting such a defense is dependent on all the circumstances of the case. This includes forensic evidence, witnesses, statements you made, your criminal history, the other guy's criminal history, the number of assailants, who was armed with what, if there is a disparity of physical strength, the location it occurred, if you provoked the encounter, why you were in the situation you were in etc. Everything affects the case and what a jury will think. Yes, there are situations where the fact you acted in self-defense is so blatantly obvious the cops don't even arrest you in the first place. But that's not guaranteed because they almost always were not there when the event went down. They have to piece it together.
In cases of brandishing and aggravated assault, your assailant is still alive and will talk the the police. He or she very likely will lie to them. Movies and TV give this impression that criminals are all despicable low-lifes that no one gives any credence. This is a harmful stereotype. Most violent encounters are between people that are at least acquainted, and they may be just as credible as you are, or more credible. This "tattooed street thug with the long rap sheet trying to rob you" most people have in there minds in the self-defense community is a rather unlikely scenario, and less likely an untrained person would be able to fend them off with only a knife.
Another thing to consider is that a knife is considered deadly force, so if you employ a knife, even if just brandishing, you had better be in danger of dying or serious maiming. I've been trying to gather cases over the years were a knife was used defensively, and I have to say they are very uncommon, and cases were the defendant was found not guilty by reason of self-defense are extremely rare. I have several cases were a person used a knife to defend themselves and killed their assailant, but were found guilty of murder 2, because the circumstances showed this was not a proper level of force to use (e.g. the attacker was unarmed and escape was available). The cases I did find where it was justified involved the defender be trapped, usually lying on there back with the attacker above them, or cases were the attacker drew his own knife first. In the later case, the defender sustained numerous wounds and nearly died, while the attacker was killed. This is part of why I consider a knife carried as the sole defensive strategy by an untrained person to be a schmuck's weapon. Defending yourself requires having more than one tool in the toolbox.
