Oh, the string weapon of infinite cool. I like this one and have spent some time musing about it. Of course my thoughts ran along the usual lines, make it a whip, put a weight on it, etc. However, most concepts ran into serious issues regarding physics or were too contrived to be usable and didn't offer much more functionality than your average sci-fi sword.
Edit: Skip the next three paragraphs if you want to get to the point faster.
Instead of whips and such, I went for something different altogether, the blinking blade. The first version was a 2 metre hollow staff where the wire is connected to the hand guard in the middle. Either end would shoot the weight out and the other end would catch it, the whole process happening faster than - you guessed it - the blink of an eye. It had a 180° cutting arc and could also be used similar to a normal staff, provided your second hand didn't grab all the way around it while you "blink". That gave it pretty good defensive capabilities too, since you could cut block a hit and cut the weapon on impact, maybe along with the one wielding it.
In practice, it was quite clunky in tight places, so the next version used an A-shape. I'ts basically the staff folded up, with the top of the A being the axis and the bridge of the A the grip. Kind of like an elbow blade that can lunge out in an almost full circle. Compared to the staff, it was much handier while offering even more range (about 1.30 metres from the hand), but it was also more dangerous to the user, and cutting a blocked weapon would most likely also sever ones own head.
I had other versions, swords that gain reach by firing the weight out of the guard and circling the wire around the tip (discarded, no point in having a blade) or just a stick that rotates it like a fan (discarded, most of the time the weight hits the target first) and other nonsense I wouldn't even dare mention.
Edit: The aforementioned point.
If you want to go for what's most reasonable though, being cool while also realistic, monofilament whips of any kind pose too many problems to be feasible, let alone the best solution. Just as in today's reality, cutting wire is much better suited for traps and netguns than for personal arms.
First off, Monomolecular wires won't last long under the stress applied in combat situations. Even if the wires would face no resistance while severing molecular bonds (don't think so), the forces needed to move them exert stress on it. Solid weapons will always have a higher resistance to wear and tear. Most implementations of monomolecular wires will be discardable one-time uses. That also means that if they are too expensive to produce, they probably won't be used at all, or very rarely and only by excentric rich people.
For example, you could have devices that are placed on the ground and fire a bunch of tiny harpoons with attached wires to instantly create a web-trap, or maybe some kind of doublebarreled gun to place tripwires (well, cutwires actually, har har) at a distance. Or fire a wire at enemies, if the projectiles carrying it have enough momentum to make it cut and not just wrap around the target with the projectiles dealing the actual blow, in which case a normal gun would do the trick just fine...
Most importantly, in a world advanced enough to have monomolecular wires, you can - or must - have all your wire weapons be controlled by embedded computer systems, making them usable without superhuman abilities. This is probably the most important part, since it is beyond human capabilities to handle something so thin that it has nearly no weight and is practically invisible.