My chicken wire hasn't been holding up well under all the snow. I'm weighing whether I should reinforce it with more poles, or if I should scrape the chicken wire and use different fencing material instead.
You could try hardware cloth which is stiffer, but the holes are slightly larger too, and it's more expensive than plain old chicken wire. I don't have any problems with chicken wire, but I live in the South where there's rarely snow.
I would see what the problem is. I agree that you just might not have enough posts to hold it up. I would start there and then decide what to use if that doesn't fix it. We have a lot of snow and our chicken wire is just fine.
I think the others are right in that you should start off by checking the posts. Be sure there is enough of them and that they are in good condition. I've seen some of that plastic type wire and it looks pretty durable.
have chickens too, the plastic wire splits in the sun and falls apart, depending on how wide your coop is you may need a T in the middle as chicken wire will flex,bend and stretch. with the post even if you have to many post if they are not in the ground deep enough they will bend out and pull to the side with heavy ice and snow if you don't shake the wire. if you sand or wet ground you may need to go deeper.