Pegs

New washing lines are so thin, best way is to buy a new line that is thicker.

A tall 2x2 of wood from B&Q or similar can be cut in a V shape at one end to make the decent props, JacBlack1, or you may find one on eBay.

I bought some soft ones in Tesco they don’t mark the clothes or slide about.

Buy wooden ones.

I still use the basic wooden pegs and don't have any trouble with them slipping. I tried some plastic ones a few years ago and they did slip.

I have the metal pegs by far my favourite and grip really firmly, don’t slide either, I’ve told my family all about them and two have also bought them,

I push them down so they grip without sliding, and my clothes ropes are narrow. i prefer the wooden ones as they grip better when pushed fully on.

I use these old fashioned, standard wooden pegs and don’t have any problem with washing slipping on the line. I do use several pegs for things like sheets/duvet covers and usually put two at either end, and nothing seems to slip. I use those soft grip plastic pegs for tee shirts, knitwear etc as these don’t leave a peg mark.

I use Minky pegs with the silicone/rubbery pads, they are brilliant. They keep the washing in place and don't leave peg marks.

www.ocado.com/products/sorbo-recycled-pegs-pack-565926011?gclid=EAIaIQobChMInaaqrt_o-AIV2t_tCh0RNAkdEAQYASABEgKb2vD_BwE&gclsrc=aw.ds

I bought these recycled ones and they were fine

I see that Minky do their own pegs, so I would assume that they are suitable for their lines. The Sure Grip ones look like a possibility.
Minky pegs

I used to have the same problem. I am afraid I don't know the brand name, but I bought these from Asda. The inner loop on the upper hole helps to grip the narrow clothes line. Hope the picture helps in your search.

That works for me too *Oopsadaisy1'

I also have some old fashioned wooden dolly pegs which narrow to nothing and keep things where you put them.

I just push them down so that the hole is not over the line.

I have some of the Lakeland non-slip ones - very good but pricey at £5.99 for 20. They have lasted at least four years now and no sign of going brittle.

www.lakeland.co.uk/4076/20-soft-grip-clothes-pegs

I bought mine from Asda or Tesco. They have rubber, or maybe silicon, and grip really well without leaving horrible peg marks.

I think Lakeland sell some that aren't supposed to slip. Don't know the price though.