The varicose veins of the leg are quite common and the ones on the back of the knee are also fairly common. It is unusual to have varicose veins of the neck. The neck veins are usually visible on lying down in some people more than in others as they become visible due to the position.. In heart failure the veins may become prominent. The only situations where the veins are visible and tortuou, when not related to position, are either due to malformations or blockage of the chest veins.