What is the value of a 1965 silver US dime?

Answer $9,000 to $20,000 at auction, IF it's a true silver error. ANACS has validated the existence of a few 1965 and 1966 dimes struck on leftover silver blanks, similar to the way a few 1943 cents were struck on bronze planchets.

Check the edge of your coin. It should show the same copper core as all other current circulation dimes, in which case it's worth only face value. If it doesn't show a copper edge it could be plated. The best way to tell is to have the coin weighed on a sensitive scale. A genuine silver dime, if it were to exist, would weigh 2.5 gm while a plated copper-nickel dime would weigh about 2.3 gm.