I've touched on this before when I went over the basic criteria you're looking for in a sword for cutting exercises but I think maybe this particular subject needs a once over. I hope to cover as many points as I can but with as little jargon as possible too.
There has always been a misconception about the katana, it's miracle steel, the whole folding process and so on, that has led to the belief that it is an indestructible sword that is superior to every other on the planet, a myth largely perpetuated by the media. The truth of it is that the sword is a tool and the process of it's construction has been changed over it's many years to best make use of the materials that were available at the time. It's not the most poetic way of saying it, but it's essentially true. Methods of producing steel have come a long way since then and a lot of these traditional methods are no longer necessary from a practical standpoint.
Take for example the folding of the steel, which produces the wonderful patterns of swirls and layers in a traditional japanese blade. This started out as a way of homogenising the content of the metal and removing unwanted impurities so that it was of even strength and durability throughout. The iron sand that was smelted down to produce the steel that was used was of such poor quality to begin with that it took extraordinary feats of smithing to even produce the raw materials required to make the swords in the first place and this folding was part of that process. Without it, the swords produced would be utterly unusable and it is a tribute to the smiths that they were able to produce such works of beauty from such humble beginnings.
It is no real effort today for a smith or a factory to simply order in the finest quality modern steel in big long bars and use that as a starting material and from a practical point of view this is no doubt the best option. The reason modern blades are folded at all is purely for the aesthetic that comes with a folded sword. It is in fact worth noting that there is definitely no advantage to using a modern folded steel blade over an unfolded one and it may come with disadvantages.
When a blade is folded, you are in essence forge welding it to itself. Every weld carries weakness and the possibility of failure. As long as you trust the forge that youre buying the sword from then you should be perfectly safe, but if you're not sure then it's best to choose an unfolded blade for cutting purposes.
Another technique that was used was something called lamination. This is where more than one type of steel is used in the construction of the billet that will be hammered out into a blade. Firstly, the smelted iron ore would be seperated into piles depending on their carbon content. The reason they did this is because high carbon steel will harden much more readily than low carbon steel. After they have done this, they can build the billet from these pieces, often placing the softer steels in the centre to add ductability and the harder pieces on the outside to make the surfaces tough and the edge resilient.
There were quite a few different layouts, the simplest being made of two different pieces, one for the spine and one for the edge and some were much more complex having as many as 7 different pieces of steel. Sometimes and usually from ebay sellers, you will find one more form of lamination known as 'maru lamination'. This means that the blade has been made from one piece of steel. so no lamination at all really. ha! Marketing speak eh?
Now the crux of this is that backyard cutters will find absolutely no difference in the functionality of one of these laminated blades which were originally designed to take advantage of the style of combat used. The key point here is the word 'combat'. We don't fight with our swords and so we dont need these additional complexities in its forging. It raises the price considerably if done properly and if done poorly, increases the number of weaker points on the sword, actually making it more prone to failure than a monosteel.
Of course, there is the last point which everyone knows about and arguably it confers very perceptible advantages. This is the so called 'clay temper' which still makes me shiver everytime I see it advertised. More accurately it should be called differential hardening. This is the process whereby clay is very carefully (in fact it's painstakingly painted on) applied in differing thicknesses to the blade and allowed to dry. It is then heated to critical temperature and quenched, usually in water. The process allows the spine of the sword to cool at a slower rate than the cutting edge and this creates two different metal structures, one significantly harder than the other. When this is done properly and carefully, it leaves you with a blade that is ductile and flexible on the back, both strong and supportive, and a much harder cutting surface that is able to take and keep an edge for longer. This is the 'hallmark' if you like, of the katana and it produces a delineating patterned line along the length of the blade, seperating the harder area from the soft. It is so coveted that if a modern sword has not been made with this treatment, many manufacturers decide to add a fake one in order to better sell their product.
For a cutter, this offers a real advantage. The edge that will hold up for longer and cut harder targets, and the support that edge needs. But this comes at a cost of course. Some flexibility is lost and a harder edge is still more prone to the damage inflicted from poor technique. I suppose at the end of the day, it is up to the practitioner to decide what is best for them.
In conclusion, for practical purposes and everyday cutting, a monosteel, high carbon, through hardened sword is probably your best bet although as you improve and you can rely on your cuts to be good enough, a clay tempered blade will give you an increase in edge durability as well as looking pretty. Folded blades are probably best avoided unless you can attest to the skill of the forge producing them. As for lamination, I can see little advantage when it comes to target cutting.
Differential hardening, lamination and folding have become a lot less necessary in these times and for most people whose toughest targets are water bottles and tatami, maybe these points are even less relevant. But for someone who has been cutting for a while, the allure of something that is made in a traditional manner can become overwhelming. I think that in a way it is a pity that the same things that held back the smiths all those years ago don't still hold us back even with our modern materials, somehow forcing the same aesthetic that made the japanese sword such an iconic piece of history. For me theres definitely something about the look of a sword that was made for functionality, yet exhibits through its construction such beauty.