http://www.college-of-arms.gov.uk/Faq.htmA coat of arms is granted to an individual and belongs uniquely to that person, being passed down from that person to the eldest son, and then to that eldest son's eldest son and so on down the male line. In such cases, the arms usually remain unchanged. While the father still lives, an eldest son can use the father's arms with what is called a "label": a small three-pointed sign which goes over the top part of the shield to indicate that he is an heir apparent. The younger sons can use another "label". A system of different coloured and shaped borders also distinguishes one member of the family from another.
In Scotland the Lord Lyon can if he so wishes take legal action against anyone who uses the Coats of Arms that belong to someone else. There is definitely no such thing as a Coat of Arms which belongs to a surname it is awarded to an individual.