The dried up plants are rather short, only about 12 inches, but they do sprawl about. The growth pattern doesn't quite look like the goats head, but the stickers surely do! My DH has been finding them in the ponies' manes, too, and I've found one or two on my VERY fuzzy malamute. Eradication is definitely in order. We're going to have to find where the ponies are getting them and probably hit it with the weed burner, if it's safe. Fortunately, the areas where the dogs go are mostly free of nasties.
