His premier as the famous bunny we know and love, was in the 1940 animated short "A Wild Hare," directed by Tex Avery. Here, he eats his crunchy carrot, as he says his first "What's up, Doc?" and the whole thing its toped by two instances of his lip smacking attack on his enemy. It seems like the letter R was eliminated from the alphabet in this animated world. Also, his performance along side Elmer Fudd sets a high standard as a classic interaction between hunter and prey in cartoons. Another thing that makes him particularly attractive as a character is that it seems almost natural that he would break the 4th wall and address his amused audience, as he does in the end of "A Wild Hare."
While he retains some of the presence he had in his first appearance in 1938, before he was grey and before his name Bugs was official, he evolved to become much more stylized and much more dimensional as a character, not to mention that he loses the overly goofy laugh he had at first. See "Porky's Hare Hunt," directed by Cal Dalton and Ben "Bugs" Hardaway for the earliest appearance of a rabbit like character, the origins of Bugs.
The rabbit made his second appearance in 1939. in "Prest-O Change-O," directed by Chuck Jones. That same year, he also appeared in "Hare-um Scare-Um," directed again by Dalton and Hardaway, with a rather crass version of Elmer Fudd (although, Bugs is still amazing, with his government refusal sticker and his brief musical stunt). In the latter, the bunny has already turned grey, although it is a great leap still between this rabbit and the wabbit featured in Chuck Jones' short.
Bugs officially gets his title in Jones' "Elmer's Pet Rabbit" released in 1941. Watch him play dead yet again, here!