I’d suspect most people relate most to the constellation that corresponds with their sign, but I’m also suspicious how many have at least one other constellation they think of as their own.
I’m a Sagittarius by birth but I also have affinity for:
Orion
Lyra
Camelopardalis
Taurus
Ursa Major (Big Dipper)
To assist in locating your own, refer to THE CONSTELLATIONS and the associated maps there.
Orion, Scorpius, Cygnus, Leo, Gemini.
I have a special affinity for Orion. I knew an Orion in high school. We were friends. Sadly, he died suddenly of a brain aneurism - no warning, no signs - just here and then gone. His parents sought me out at his funeral and said “You’re the girl in the picture on his wall!” Apparently, although I’d never known it, he had a bit of crush on me and had blown up a photo he’d taken of me and hung it on his wall. I always look for Orion in the sky and smile.
The Southern Cross, Orion, Cassiopeia.
The ones I can pick out immediately:
Ursa Major
Orion
Cassiopeia
Pleiades
Big fan of the Dippers, spending the summers of my youth on the beach in MI and seeing them up there at night; campfires on the beach being one of my all-time favorite things. Cassiopeia, too, same association; it’s just really easy to identify, being a big W in the N/NE at the same time.
And then Orion. Had to do an art project in middle school and somewhere saw a simple drawing, which I carved clumsily into a rubber pad to make a block print of. After that I noticed Orion in the sky in the winter evenings, my college years drinking buddy. OK I didn’t really drink much, but I do remember seeing him in the sky on my walks home at night.
I was walking my dog with my niece, who has a jewelry business named Bellatrix, so I pointed out the red star in Orion, assuming she knew it. Nope, she named it after the character in Harry Potter. Odd that a cute blond would use a nasty brunette, but there you go. She enjoyed learning it was a star and where it was. 
Also walking my dog on winter nights I’ve been noticing Orion’s neighbors: Taurus (my wife is a Taurus), Gemini (Castor and Pollux are easy to spot), and Canis Major (Sirius, the brightest star in the sky!)
And most recently I noticed the Northern Cross, part of Cygnus (IIRC). I also remember the Southern Cross from my (long ago) sojourn Down Under, which is definitely remarkable, as noticeable in the southern sky as the Big Dipper is here. I found I could use it and Orion to help find South, though I don’t remember quite how!
I’m a Leo but have no idea where it is, what it looks like, or when to spot it.
Of course! Forgot that one.
I live in Durham, but even before we lived here, my wife had a business called Seven Sisters Designs, as she’s one of seven sisters. Pure coincidence that we moved here where it’s the town’s hallmark (well, after The Bull, that is). No idea why Durham picked Pleiades as their logo.
Nitpick: The Pleiades is a star cluster, not a constellation. It forms the shoulder of Taurus the Bull.
The Hyades is another, open star cluster, and it forms Taurus’s face.
Orion, especially after reading Robert F. Young’s “Written in the Stars.” 
Which postulates that the shape of the constellation spells out “fuck” in an alien language.
Argo Navis
(Screw the IAU)
Orion.
When in Iraq my best friend had to fill-in for my flight engineer when we had to go to Mosul. It was really dark on the airfield so I was looking up (I grew up in the city and I’ve always been amazed by stars). He sees me, points up, and says, “That’s Orion.”
“How do you know that?”
“Because I grew up in the country, mother fucker.”
He was killed about three years later in Afghanistan. When my wife and I have a son we’ll name our son after him, obviously. We might do the middle name of “Orion.”
I would be willing to bet that after The Sun, The Moon and maybe Mars, Venus, Jupiter or Saturn, the “heavenly body” most people could identify would be Orion, The Big Dipper or (with any luck) The Great Square of Pegasus. But surely Orion!
ETA: Or on a good night, The Milky Way.
For me it’s Ursa Majorthe Big Dipper. It brings a fond memory of my grandmother, who got a college degree in her fifties. She took astronomy for a science requirement, but already knew all the planets, constellations, and such.
Grandma showed me how you can tell the time of year by seeing UM/BD. On October 15 it is at the “bottom” of it’s counter-clockwise circling through the northern sky. So around mid April it’s completely upside down. Right about NOW it’s at 0900, if you visualize it as traveling on a clock face.
I think is a bit better suited for IMHO.
I have written a series of unpublished (perhaps unpublishable) fantasy novels about an “avatar” (so to speak) of the constellation Lyra. Her name, not indicative of any great imaginative prowess, is Lyra.
Absurd, but there it is: Lyra is my favorite constellation.
(A Sunday School teacher once told my sister, “When we die, we go up into heaven and become stars.” My sister said, “I want to be a constellation.” So, when that happens, she will be my new favorite.)
Leo’s a springtime constellation. In most of the US, it’s about halfway between the horizon and zenith when it’s due south.
http://stars.astro.illinois.edu/sow/cm4.html
If you’re going by astrological signs, it’s no longer where it was 2000 years ago. The Sun doesn’t rise in it when you would expect any more.
Ursa Minor, for several reasons:
What a timely thread. I was just talking with my son about constellations and our zodiac signs and how astrology is complete garbage.
The constellations I have a fondness for are Orion and Ursa Major and Minor because they were the first ones that I can remember being able to find on my own.
Back in the '60’s a couple of friends and I used my new copy of A Field Guide to Stars and Planets (Peterson Field Guides) (which was older than the one in the link but which had some great charts) to learn all the constellations visible in Nashville over the course of a few nights in the summer. With the exception of the hours just after sunset and just before dawn we learned all but about 3 hours worth of sky.
One of our jokes was to see two stars and say to each other, “You see those two stars that look like a Buick?” (or some nonsense like that.) Later on I got a celestial globe to learn the rest of them, including the Southern Hemisphere. I doubt I could names more than 50 now (of the 88.)
Another neat book I picked up later is Star Names: Their Lore and Meaning By: Richard H. Allen
The red star in Orion is Betelgeuse. Bellatrix is blue, and is the other shoulder.
To the OP, most people don’t even know their own birth sign (or rather, they know it incorrectly), many of them don’t know that the birth signs correspond to constellations, and even of those who do, most of them don’t know the constellation of either their real or purported birth sign. So I really doubt that birth signs dominate the list of favorite constellations.
Personally, I’m particularly fond of three:
Orion, for the same reasons as most others, plus the presence of a nice naked-eye Messier object, and an easy indicator for both another Messier and the brightest star in the night sky.
Sagittarius, because you can pretty much just point a scope (or even just binoculars) at a random spot in that direction and see some deep-sky object or another, including both the Lagoon and Triffid Nebulas. Plus, of course, Sag A*, the black hole at the center of our Galaxy, and the brightest radio object in the sky.
And finally, Delphinus. It’s one of the few that actually does vaguely resemble what it’s supposed to, but at the same time, it’s dim enough and obscure enough that it’s really hard to find, which makes it seem like a sort of private secret of mine.