The type of steel preferred is going to be highly personal. Outside Japan, different smiths hand making swords in the japanese style can have their own personal preferences for which exat grades of steel they prefer to use based on their personal forging methods, although tool grade steels like L6 and 1086 seem to be the most commonly used grades. Unless the blade is made from tamahagane steel produced in a traditional tatara smelter, folding doesn't do anything to make the steel any stronger. Folding is highly desirable these days because of the visual patterns it creates in the steel, although badly done folding can actually weaken the steel by creating internal gaps in the steel where the folds have failed to weld together properly.
In the case of tamahagane, folding does help because tamahagane is inconsistant in quality straight out of the tatara smelter, and the steel is broken up into small pieces, so the folding process welds the pieces together into a block large enough to make a sword blade from, and improves the overall quality of the steel (giving it a more even carbon distribution, removing gaps and voids in the steel produced by gasses during the smelting, etc). Tamahagane is not bright and shiney like modern steel, it's rough, has a dull colour and can actually look like stone before the smith starts working on it.
With modern steels, the steel mills can have very precise control over the quality of the finished steel. Swedish powdered steel is particularly good quality because it's produced in a way that keeps impurities to a very low percentage of the steel (the steel is smelted inside sealed vats). An unfolded sword blade made from high quality modern steel can be very good, but the type of steel used isn't the only factor involved.
How the steel is heat treated to harden it after it's been shaped into a blade is as important as the quality of the steel used in the first place. High grade steel that has been poorly heat treated can make the blade either too hard (it will chip and crack too easily), or too soft (it will bend too easily). For a knife, the heat treatment isn't as critical as it is for a sword, because of the length of the blade (a good heat treatment is important for a knife, but it's even more important for a sword). A sword blade is expected to flex a little bit when cutting, but should return to it's normal straight shape (just like the springs in a car's suspension will flex over bumps, but return to their normal shape afterwards).
For a European sword, a "spring temper" is ideal. This involves hardening the sword by heating it up and keeping it at a high temperature for a period of time, then quenching it quickly in water or oil, which hardens the steel, and then reheating it a second time to a lower temperature and letting it cool more slowly to re-soften the steel a little bit, giving the blade it's desired springyness. The Europeans tended to use "case hardening" where the entire outer skin of the blade was made harder to hold a cutting edge, with the inner core of the blade softer to withstand the stresses of cutting.
The Japanese did (and still do) something similar, but in a different way. After the blade was shaped, the smith would coat the blade with a mixture of clay and ash from the forge fire, with a thin layer near the edge and thicker layers on the sides and back. Then the smith would heat up the blade to a high temperature and quench it, like the European smiths did. The difference is the clay coating. During the quench, the clay acts as a type of insulation, keeping the heat in. The thin layer of clay at the edge looses heat faster, so the steel at the edge sets first and is harder. The thicker layers of clay on the sides and back loose heat more slowly, so the steel in those parts of the blade set more slowly and are softer than the edge. This is called "differential hardening", different parts of the blade are given different hardnesses in the one process. On top of this, Japanese smiths usually assembled the blade from several different pieces of steel with different levels of carbon. Higher carbon steel was used for the edge, lower carbon steel for the inner core, sides and back. This differential hardening also has a visual effect on the steel. After polishing, the edge will have a different visual appearence to the sides and back of the blade. This different appearence is called the hamon, or "temper line" as some people call it. This line is a visible boundary between the harder edge steel and the softer side steel. A skilled smith can affect the shape and pattern of the hamon by putting the clay onto the blade in a particular pattern before heat treating it, and since the Japanese are obessed with naming things, different patterns for the hamon each have their own names.
A very good web site with much more information about Japanese swords is:
Rich Stein's nihonto site
