It is more common to write: "People care too much about their appearance." This is because it is being used as a "non count" noun and the plural could confuse people into thinking the writer means the occasion when someone is in a play, movie, etc. So with our would you still use singular?
Note that these dictionaries note it as being singular or plural (appearance as looks): http://www.macmillandictionary.com/dictionary/british/appearance http://www.oxfordlearnersdictionaries.com/definition/english/appearance
Anyone shed light?