I have two lungelines.
One is the flat nylon - it's very light, very smooth AND I hate it - never, ever use one of those without gloves
Also, if you are lungeing outside on a windy day the lightness of the nylon has the line flapping crazily in the wind.
The second is the flat padded cotton - it's a bit heavier than the nylon but I like the feel of the padded 'stops' (makes it very easy to let out a little line at a time) and, I think, helps if you happen to have a horse that pulls. And, although one should always wear gloves when lungeing, this one is more forgiving if you happen to forget the gloves
I also use a lungeline attachment that clips to each side of the bit so that you don't have to change the configuration of the line when you change sides.