As an adverb or adjective, upward means "toward a higher amount, degree, place, level, or position": "The stock market moved upward." "Today, the market is having an upward turn." Upwards should not be used in either of these sentences. The phrases "upward of and "upwards of are standard, but only in their proper meaning of "more than," "in excess of a given quantity." Neither phrase should be used to mean "less than," "about," "almost," or "approximately." The statement "His accident cost him upwards of $500" means that it cost him more than that amount.
Dictionary of problem words and expressions. Harry Shaw. 1975.