If you wish to follow the rule, then I believe that "appear" is correct. In "there" sentences, we are told to ignore "there" in analysis. So we have: "____ to be no survivors." Next, we put the sentence into regular, non-question order: No survivors ____ to be. Thus, we need "appear" because "survivorS" is plural. (No survivors appear to be. = No survivors appear to exist.) ***** Compare: There appears to be a big problem between country A and country B. (A big PROBLEM between country A and country B appearS to be.)