As a young "almost ready to drive" when these three cars were new (actually I was in Brazil at the the time, but faithfully went to the airport magazine shop every month to buy my copies of Popular Mechanics ), I can say that in the minds of us young ones the Eldorado was the clear winner. 1956 was the year left their more conservative ideas of luxury behind and went with pure "flash and dash." For instance the '55 Cadillac 62s and Coupe de Villes were still pretty plain compared with what came the next year. The colors offered and the interior materials were really flashy compared to those offered before that year.
My aunt and uncle had a 1955 Eldorado (black with a red and white leather medallioned interior), basically a model 62 convertible with a bit=it more chrome, fancier interior, wire wheels and a textured metal lower rear quarter panel. Not a bad car but not really up to the 1953. The '56 was the year they upped the ante with the big, distinctive tail fins. I never drove a 1956 Eldorado's, but I spent plenty of time behind the wheel of a '56 Fleetwood my dad had. From a young guy's standpoint the Cadillac was the clear choice among the three featured hear.
I never drove, or for that matter saw too many, Packard Caribbeans. One of my best friend's dad had a a 1955 Packard Patrician, a nice car but very conservative (despite the engine turned dash appliqué) and while the self leveling rear suspension was just not up to Cadillac's design and take on luxury.
The Lincoln, which even then I considered really beautiful, was a real also ran in the prestige games of the luxury class, and to be honest was rightfully and overwhelmingly overshadowed by the magnificent Continental Mark II. The following year Lincoln completely blew it by tacking monstrosity tail fins on their basic beautiful design. I never drove a 1956 or 57 Lincoln but I did get to drive my aunt and uncle's Mark II (it replaced the '54 Eldorado when my aunt, a leading fashion designer was offered one of the first ones in 1956). The Mark II was originally offered for sale to high profile (reputable), what we would call first rung celebrities today. To this day the Continental Mark II remains one of my "all time" favorite post war cars.1957 was the year that Detroit really through out their old ideas and the big three automakers really let it all hang out with their flashiest and possibly best designs of the post war era. The cars not only looked different but because of more powerful engines, lower center of gravities and new 15 inch size wheels handled a lot better too! The difference between the '56 Fleetwood and our 1957 Olds Ninety-Eight four door Holiday (hardtop convertible in the vernacular of the day) was like night and day. The Olds was a bright, fun car that let light and the fresh air in, while the Cadillac was a dark gloomy (regardless of exterior color) old farts car (I never used it on dates and really only drove when my dad asked me too. The Olds
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Today all three of those cars look great, but they don't have to really compete in a real marketplace anymore. If I were to choose one today it would, hands down, the Lincoln. Its' design combine great style with good taste, some which in all honestly I can't say for the Eldorado or with its' three tone paint job which really made it come off as an old women with too much makeup (then and even now). The Eldorado was Las Vegas cheap flashy and found a home (particularly a second home) in the garages or apartment parking lots of those for whom a car was their biggest investment!