What year was silver removed from US dimes and nickels?

The question is actually "which years" rather than "which year".

Nickels:

Contrary to popular misunderstanding, nearly all US nickels are made of an alloy of 25% nickel and 75% copper. The ONLY nickels that ever contained silver were special "war nickels" minted from late 1942 to 1945. War nickels were made of an alloy of 56% copper, 35% silver, and 9% manganese because nickel metal was needed for the war effort. These coins are distinguished by a large mint mark letter over the dome of Monticello on the back.

Dimes and quarters:

1964 was the last year for silver dimes and quarters. Starting in 1965, those two denominations were changed to the current cupronickel-clad composition.

Half dollars:

Until 1964, halves were made of the same 90% silver alloy as dimes and quarters. From 1965 to 1969 they were made of 40% silver. None were minted for circulation in 1970. Starting in 1971, they were changed to the same clad metal used for dimes and quarters.