Well, I'm not really a World War II specialist, but I think I can take this one on. The Navy actually didn't fly a large number of different aircraft types off carriers during the war, so the answer is relatively simple.
At the time of Pearl Harbor, most squadrons were equipped with three types of aircraft. The TBD Devastator had wings that folded vertically. The SBD Dauntless and the F4F-3 Wildcat had fixed wings.
I believe one or two carrier-based fighter squadrons were still flying F2B Buffaloes in early 1942. The F2B's wings were fixed.
Shortly before the Battle of Midway the fixed-wing F4F was replaced by a later version with folding wings. The F4F-4 (and, I believe, the later versions of the F4F-3) had outer wing panels that twisted and pivoted backward, so the wingtips wound up next to the horizontal stabilizers.
Shortly after Midway the TBD was replaced by the TBF/TBM Avenger, whose wings folded backward like those of the late F4F.
The F4F was eventually replaced by the F6F Hellcat, whose wings also folded backward.
The SBD was replaced by the SB2C Helldiver, whose wings folded vertically.
Some carriers carried the F4U Corsair, whose wings folded vertically.
I believe those were the only combat types flown from carriers during the war. In the first few months some of them carried a few SNJ trainers and J2F Ducks as "utility aircraft"; both of those types had fixed wings. At the time of Pearl Harbor a few old SBC Helldiver biplanes were still on board American carriers (there are color photos of them on board the Hornet, which wasn't commissioned at the time the war started); they had fixed wings too. And, of course, for the famous "Doolittle raid" the Hornet carried 16 Army B-25 twin-engined bombers. And various other Army types (and a few British ones) were carried by American carriers on ferry missions.
Some of the WWII experts may be able to think of some others, but I think the above comes close to covering the subject.