Sounds like an illuminating project. They make so many cute designs in these lights now that they are hard to resist. I have some dragonfly ones and don't have them up yet, but my thoughts are that you could use cup hooks very easily. A drawback would be if it's very windy through that area of your yard, they may work loose off of the curve. Another options, cheap and not too permanent (as in holes drilled or anything) would be zipties. I love these. They are available in every color and size you can imagine, so perhaps it wouldn't be an eyesore.
Have fun and good luck.
Jan

Go to your hardware store and ask for nail hooks that are designed for this purpose. I don't know the brand name etc but they have a piece where you put the nail in and a hook piece off of that. You put the cord in the hook piece then nail it all to the wall/trellis etc. The nail is not very big but it does the job! Sorry but don't know how else to describe them!
We use them every year to hang our lights on our fence posts and other places and will be using them this weekend to hang strings of clear christmas lights on a piece of lattice.
Will see if I can attach a pix I have just taken of one in the computer room I have up. [Well figured out you can't post pix to this forum so will email it over!]

I used a staple gun the last time I did this. I, obviously made sure it wasn't plugged in. You have to be very careful to line up the staples over the wire so it doesn't pierce the wire but it is possible.

They make cup hooks with a 'clip' over the opening.

Nonacook:
Do you find that type of cuphooks in the hardware store?

I built this trellis last spring using some discarded display racks that I got from work.
I intertwined the lights from top to bottom, side to side before I planted the jasmine vine.
I wish I had a pic of this with the lights on.

Here is a link that might be useful: {{gwi:1539}}

Please be very careful if you are using anything metal to hang them up with or hang them onto i.e. staples, hooks etc.
My oh reminded me of this when we found a couple of boxes of lights in the cupboard and I decided to put them around the pond. I said I wanted to find some small shepherds hooks to use - until he reminded me about metal being a conductor.
BTW, he spent 25+years in Federal Civil Service in a job where electrical safety was paramount.

Yes, It does say on the boxes of the twinkly(tiny christmas tree) lights not to use anything mettle. that disclaimer being said, I have used cup hooks, they can come with a sort of rubber coating. Then to attach them to my bamboo privacy screen fence that is mounted on top of chain link I have used the twist ties out of some boxes of "Baggie" style bags, if you use the ones out of the lawn and leaf bags they can be cut in half. They are covered with a surface like paper or thin plastic. I know that its risky, but I felt safer doing this than the staple gun method. I am really sure that it says not to do this on the pack, DISCLAIMER!!!!!! but..... It does work, and its easy to go up, and also to come down, as they eventually get decrepit. I have also used wire, Of course you would want to get the fine little plastic coated wire they use for electrical wiring. In a pinch, and this really works well because its the exact right length, I have used Christmas tree ornament hangers. YES! they are not coated, and there for, You should never use them with anything like plug in lights!....unless in a fit of needing to get something done you are willing to live dangerously........

I am pretty sure those nail thingys with the hook is for telephone or speaker cords. The hook thing is made of plastic. You can get these at your local hardware store or any department store with a hardware section. Might even find them in the section where phone cords are sold.