Dominic Frigate Bird - 6 May, 1962.
Frigate Bird was the first and only American full-up nuclear missile test, but the Soviets were way ahead with.
The original post was asking about full tests with bursts "...on or near the earth's surface."This would exclude the Hardtack and Fishbowl tests, which were (as all here know) mid-to-high altitude air bursts.Edited: To clarify.