The line chosen depends on many variables: the style of the necklace is only one of them. A very important one, indeed.

I use chains and wires of various metals (silver, gold, steel, etc) leather thongs, different textile threads (cotton and hemp my favorites, either single or several threads together, braided or not).
Depending on the project, industrial materials may be of use as well: electrical plastic-coated copper wire (why not?),(not-nylon) fishing line*, etc.
Sometimes it may be advisable to use different types of line in different places of the same necklace -- i.e. springly wire to string the beads, with a leather thong or a vegetable fiber braid tied at the ends of it, to finish the back of the necklace with a sliding device, in the traditional size-adjustable Indian style.**
*Beware, some of these, specially those too resilient, may be dangerous: wires, or cords used in ships, could seriously cut flesh in the event of the necklace getting hooked somewhere. A "security point" should be there, for the thing to break just in case.
**I could show several cases/images with different solutions chosen, if required.

But of course, more than anything else, the critical point is the size(s) of perforations. There are many tricks to reduce the gauge of a perforation, the problem is generally the need to enlarge it, and the possibility or convenience of doing this: i.e. when you do not want, or dare, to change an ancient bead.

In the necklace shown here the case is rather complex: the lapis beads have a confortable 2 mm gauge hole; the modern gold-and-ruby discs have a 1,3 mm perforation, which could be enlarged to some extent if needed; the ruby beads have a small hole and, although very hard, could also be enlarged at will; now, the tiny old Basra pearls are extremely fragile and their perforations are not much larger that 0,3 mmm.
Not easy to decide.
So this necklace is strung on nylon coated stainless steel wire (0,3 mm), which I consider to be for cultural and practical reasons (it bends and stays at an angle, or breaks suddenly after some time) only a temporary emergency expedient...although perhaps it is less temporary than hoped for, as happens all the time in society at large.

From 2011:

http://beadcollector.net/cgi-bin/anyboard.cgi?fvp=/openforum/&cmd=iYz&aK=77353&iZz=77353&gV=0&kQz=&aO=1&iWz=0

http://beadcollector.net/cgi-bin/anyboard.cgi?fvp=/openforum/&cmd=iYz&aK=57881&iZz=57881&gV=0&kQz=&aO=1&iWz=0

When I wrote nylon("not-nylon fishing line") I meant something that probably goes for another name, I meant the bluish, clear line that almost everybody uses for fishing, and looks like silicon, and is everywhere.
Well, as you can see, I am not very technical nor fastidious about my ways...not serious. My father told me more than once: "You are a butterfly".
Not an excuse, Mr. Freud.
But I am happy my post has been such a good opportunity for our revered Beadman to refresh our memories and enliven thus our beloved Forum.

Among "sporting goods" there are lots of different lines that are used for catching fish or sea life.

But the most-typical thing called "fishing line" is better described as monofilament. It is very strong extruded plastic. It's a good line for bead storage--but not necklace-making. Eventually it will become brittle and will break. Plus it does not have good "drape."

Structured lines are much better for strength and drape.

Lines can be twisted-together, or braided together. Nowadays that are lots of very fine strong braided lines.

At Facebook, one of my friends recently told me about an exceedingly fine fishing line call "Frog's eyelash." Imagine how fine that must be!

Right, what I was hinting at when saying "nylon" was precisely that: extruded plastic.
Right, good for bead storage (and also fishing I guess), not necklace-making.
Now, "Frog's eyelash": whoever christianized the product, I deem to be a genius of merchandising, or literature, or both! And I should look for it(*), no doubt.
Thank you, Beadman.

(*Well, a Google search had unexpected results).

This is a screenshot of the exchange at my FB Group, Beadman - My Stuff.

The funny thing is: probably the lines we use will out-live the necklaces we create with them.

See here a line that would be OK on many projects:
I found it on the street, somebody lost(and I salvaged) it, the connecting wire from a Sony mobile phone to its ear plugs.
It does not break or age quickly, its thickness is adequate to most beads' perforation, and its length is enough for most necklaces --even for two chokers...
And FREE!

Please do not accuse me of being a loose man, I know how to suddenly turn purist if needed. Here the proof to redime myself:
In the first one (Indian openwork with Venetian arabesque beads) you could believe it is an antique traditional Indian adjustable string. I went as far as rubbing the cords with oily hands (mustard cooking oil, then with some sandal oil to finish).
The second example is a Sind wedding necklace of which only the loose beads came to me. The beads are of course antique, but at the back, salvaged old Indian threads were used. And it smells ethnically, too!
Labor costly.

This is part of a discussion I initiated at my Facebook Group. It was about non-gem-quality "ruby" beads from India, and what they might really be. Lots of opinions were offered (!). This is one of the final photos.

In case a second try is more successful....

This is the reason left overs are the offerings sacred to lord Shiva: he is the Clever One.
I have many derelicta at home, like the hand-washing bassin, a street surface-find!
And recently I read that "gazpacho" (our world-famous Spanish recipe) is an ancient Iberian word, meaning precisely "left overs".

You stirred a memory in my brain, of the famous Mahatma Gandhi's words:
"What has value has no price".

His hobby was making LINE: he spent all his FREE time at the spinning wheel!

My length of line,
I would wrap around myself
till a cocoon were made.
Then, at the season right,
would I break free of it--and fly
as a dreamed butterfly
in search for the perfect bead...

(The bead I am the hole of).

I'm sure there are many ways to think about and describe transcendence.

This is a good one.

J.

This necklace was my Christmas-cum-Birthday present to my sister.
While at her pleasurable duties as a grandmother, her 6 months old grand daughter took hold of it with such gusto that the thin silk line broke. By chance all the beads except one were salvaged, and it was quickly restrung (using this time a plastic coated steel line).
As one of my friends had commented that perhaps the three pendants were set too close together, I saw here/now an opportunity to try the new option and liked it. A small change and so different looking! A learning experience: learning to listen to other people's comments.
See it below.