Knives need more training than firearms to be effective in SD situations. One of the biggest issues with knives is that females carry them, think they are safe, and are easily disarmed because they are overpowered and don't know how to use them. I've even seen females carrying the "wrong" type of knives (box cutters, flat edged 2-inch buck knives etc). The "right" type of knife is built for the purpose of maiming, has serrations (interesting bit, knives with serrations are much better stabbers because a flat edged knife will create a vacuum in the body and be extremely hard to pull out after the first stab), is fixed, and is as long as can be reasonably carried. Obviously laws get in the way of those last two, so most SD knives end up being 3-4" folders with serrations. Grips/handles are equally important.
I took a HTH course with a former Israeli Drill Sergeant. Part of the course was unarmed knife defense. He imparted some wisdom when it comes to knife attacks. There are only two types: HTH combat in a formal military setting, and crimes of passion. Crimes of passion is what we would focus on here. When you see a stabbing on the news, it's never the victim was stabbed three times, or 5 times, or even 10 times. They are always stabbed dozens of times. This is because if someone is stabbing someone else, it's because they are mad, heartbroken, drugged up, etc beyond reason. As been alluded to by other posters, knife crime is almost always extremely brutal and violent. But we can use this to our advantage. The biggest advantage is that rarely do people die from one stab. This takes time, is very noisy, and is usually something a person with a little knowledge can fight back against. In fact, this former IDF Sergeant claimed, it takes, on average, seven stabs to kill someone. Remember, this is a crime of passion, not a trained operative assassinating someone in the dead of night with a Rambo knife. They will not know where to stab (they probably won't care), how to stab, or how to defend should they meet resistance. They may try to use a non-SD knife in the attack, further limiting their effectiveness. Defending unarmed against a knife attack is surprisingly simple when you keep in mind those factors. But we aren't talking about a knife defense, we are talking about a knife attack.
I carry a knife, specifically a Leatherman Skeletool, with serrations. I would never dream of using it in self defense except as a very very last resort.
SD Knife classes are available. I know Glenn knows some people who do them, or at least I think I remember him posting something about it a while back. These classes are a necessity should anyone decide to carry a knife in self defense because of the intricacies and "personality" of knife combat.