Knives can be protected by the Second Amendment. Remember, the Second Amendment is for 'arms', not 'firearms'. According to District of Columbia v Heller (hereafter referred to as 'Heller') any arm sufficient for a militia is protectable under the Second Amendment, but that standard isn't necessary. Self-defense and hunting are examples of other things that can extend protection of the Second Amendment. Scholars (such as David Koep, Clayton Cramer and Joseph Olson) argue that under the Heller standard, knives are clearly protectable, so long as they are knives that are arms and not necessarily just tools.

Under Heller the Supreme Court pointed out that the Second Amendment includes 'all instruments that constitute bearable arms', and they defined arms as 'any thing that a man takes into his hands ... to cast at or strike another'. But excluded are 'weapons not typically possessed by law-abiding citizens for lawful purposes, such as short-barreled shotguns', so not necessarily everything is going to be protectable. If it's a weapon that people typically wouldn't have for lawful purposes, or if it isn't something that's typically conceived of as a weapon (i.e. a kitchen knife) then it likely won't be protected.

This is supported by subsequent state-level decisions like City of Seattle v Evans (Washington State) where it was ruled that a paring knife is not an 'arm', so it doesn't receive second amendment protection. Mr. Evans had been charged with unlawful use of weapons for possession of a fixed-blade knife, after an encounter with police where they located a concealed paring knife. In case you're not familiar with paring knives, they're a small type of kitchen knife used for cutting apples, deveining shrimp, etc.

The Washington Supreme Court ruled that the paring knife was 'a utility tool, not a weapon' and that 'not all knives are constitutionally protected arms'. The Court went on to detail in their judgment that they could not possibly include kitchen utensils as protectable arms under the self-defense standard from Heller because it would mean 'there would be no end to the extent of utensils arguably constitutionally protected.' They point out that while admittedly any sharp or hard object could be used as a weapon, that doesn't make them arms, because it would make a vague and unworkable law that would lead to virtually everything being protected. If a kitchen knife constituted an 'arm', then so could rolling pins, frying pans, rakes, bricks, and other things - and those are all real examples the Court gave us.

So just because something CAN be used as an arm/weapon, it doesn't make it protected. Something that isn't designed to be used that way most likely won't be protected. What about swords? As a second amendment issue the distinction between knives/swords isn't really relevant. As you can tell from the above, relevant distinctions are more about what the items are designed for, how likely it is that a lawful citizen would have one for a lawful purpose, and other more technical things like that. The above analysis applies equally to swords as it does to knives.

Something like a knife easily fits with the Heller definition of arms, so long as it's a knife designed for use as a weapon. However, these types of cases rarely get far. So it isn't as clear as it is with guns.

In terms of the types of knives that states ban or heavily regulate - switchblades, automatic knives, and the like - there isn't really a great deal of Supreme Court precedent on it. There are a few state-level cases that can give us some guidance though, such as State v Delgado (Oregon Supreme Court), where a ban on the manufacture, sale, carry, etc of switchblades was struck down for being in violation of the 'right to bear arms' section of the Oregon State constitution. The Court ruled that the term 'arms' did not mean 'firearms', but it meant virtually any hand-carried weapon commonly used for self defense. This was basically an extension of a previous decision from the Oregon Supreme Court, State v Kessler, which ruled that the possession of billy club's inside one's home was protected under the 'right to bear arms' section of the Oregon Constitution. It's possible this same language could extend to the US Constitution, but that isn't clear, so we must remember that when considering these Oregon cases.

The State government, which did not win in either case, argued that switchblades were not typically possessed for lawful purposes, and were almost exclusively things possessed by criminals for the purposes of committing crime. If they could prove this nowadays in the post-Heller world, they would have a pretty good chance at winning. And a post-Heller case in Indiana did uphold a ban on the possession of switchblades in the case of Lacy v State in the Court of Appeals. This does not bode well for the Oregon State v Delgado precedent. However, there are many states where switchblades are not banned, so one could argue that they can be and are possessed by law-abiding citizens for legitimate purposes (self-defense, most likely).

Note that that Kessler case was about billy clubs, an area with even less coverage. So it's possible under the right circumstances that some types of clubs, batons, etc could be protectable.

So yes, there is some Second Amendment case law, but I would say it isn't really a settled area of law at all. I hope the above helps!