Cadillac's fins began as modest lumps at the trailing part of the rear fenders, but in 1956-57 Chrysler started a brief, intense fad for large fins of varying shapes that eventually assumed rococo grandeur. Cadillac stylists (well, perhaps more marketing staff than the stylists, though the about-to-be-retired styling supremo Harley Earl had to have agreed) were swept up in the craze and created the most extreme, outrageous Cadillac fins for 1959.
Sanity began to return to the American automobile industry as 1960 approached. Bill Mitchell, Earl's successor, favored crisp, well-tailored styling and began the process of de-finning Cadillac for the 1960 model year when he finally was able to affect what was in the production development pipeline.
This photo shows the apex of Cadillac tail fin extremism, a reply to what Chrysler successfully offered in 1957.
Cadillac was stuck with the same bloated body it had in '59, but the fins were lowered and simplified with the elimination of the tail light housings and some minor reshaping.
This model year marked a restyled body. Tail fins were flattened and slightly lowered. However, an echo or counter-fin was added to the lower part of the rear fender zone. This helped to make the car less heavy-looking.
Cadillac styling was almost unchanged for 1962.
1963 brought a revised body: note the straighter A-pillar and cleaned-up sides. The lower "fins" are gone, but the tail fins are essentially unchanged.
Tail fins might be slightly lower for 1964.
Now the fins are gone, as seen in this brochure photo that looks like it might have been taken at GM's Tech Center where corporate research and styling were located.