It's singular because the number before the noun is "one", but if you were to change the order it would be plural: "He gives his three children one, five and ten dollars respectively".
I'd argue that it's got to be, "ten, five and one dollars" as listing numbers that way is an old form of describe an amount. As in, "Four score and seven years ago […]," which describes the numerical value 87 years. So if it were1, "Four score and one years ago," which describes the value 81, it wouldn't really make sense to use the singular, would it? I could be w