First appearance in a story: Detective Comics #40 (cover-dated June 1940),
though he was seen in the last panel of the last page of the Batman (and Robin) story in Detective Comics #39:
And from Detective Comics #40,
He didn't originally have the power to physically morph his body, but that would happen many years after his original appearance.
By the way, was that Clayface reforming in this panel from Infinite Frontier #0?
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