When choosing a mover ask if he carries Workers' Compensation Insurance. Such coverage is required by the DOT. This adds to the cost of doing business but it protects you, so make sure you only choose a company that is covered.
When choosing a mover check his BBB record. Local Better Business Bureau offices keep records on companies registered in the state. Moving company that has more then few unresolved complaints should be avoided. More information on household storage here.
When choosing a mover ask if it is a member of the state moving and storage association in the state in which they are based or a member of the American Moving & Storage Association (AMSA). A reputable company should be a member of one or the other.
When choosing a mover make sure their place of business actually exists. Beware of companies who provide no business address on their website, business documents, or a company that is hesitant to provide you with their physical business address.
