Depends on the sword and the era. Modern fencing foils can bend almost into a U shape -- but you have to straighten them out if it gets too bad (you want a slight bend). These are designed for sport; if the blades were thicker they would puncture instead of bend and running someone through can ruin your match even with electronic scoring and all that stuff. The foil blade is basically thin and square.

Epees, which are rather triangular in their blades, are heavier and do not bend as readily as a foil. That's as it should be, because epees evolved from the rapier and small sword. In foil, the torso is the targat, in epee it's the whole body (front and back in all cases).

A sport saber has a rectangular blade and is descended from the slashing sabers used by cavalry and dragoons. It was intended to be used as a cutting instrument (although you can use the point as well). In sport saber the torso and the arms are targets.

ALL sport weapons are blunt on edge and point, and the points are bated with rubber or plastic tips. There is A LOT of safety stuff in all sorts of fencing but especially in sport fencing.

A true rapier might have a fairly flexible blade, but it might not. Depends on the school and preferences of the owner. The French, the Italians, the Germans, the Spanish, the Hungarians and for all I know the Estonians all had their own schools of the sword. Mostly you don't want your blade to become easily nicked (it'll break first there) or to be so flexible it can't pierce heavy clothing, etc.

Getting to other swords, like the backsword (the so-called "claymores" of the Highlanders), these can be used for both cut and thrust. Tough blades, only barely flexible.

The saber made for use reached it's peak with the 1913 Patton saber (George Patton, the General, was an Olympic fencer). This was just in time for it to be tossed out as a military weapon, but it's still probably the best one ever designed. The saber originated (as much as you can say things like that) from the Hungarian saber, which of course has a lineage back through the Muslim swords, and...well, you get the idea. Anyway, it was intended to be used to chop at people from horseback, whether those people were also on horses or on the ground. The blade was heavy, as you'd expect a meat cleaver to be.