Obviously, a lot of the unique weapons are obtained through ways you'd only come across in your own game, so this thread has the potential to include spoilers to your adventures.

I've got 3 unique weapons, thus far (but I'm only 11 hours into the game at the moment).

Chopper, Big Boomer and That Gun.

The Chopper is basically just lying around at a certain farmhouse location, only picked it up when I noticed it's valuation. Big Boomer I collected after inadvertantly shooting the owner. It's a shotgun worth around 2500 caps. I loved the Terrible Shotgun on F3, so this should be right up my street. However, I'm actually contemplating restarting from prior to that as the owner in question has a few quests I'd like to do, first. That Gun is for sale in Novac, and again is a great weapon, although the reload time on it is poor. My usual weapon of choice is either the Mercenary Grenade Launcher (as part of my bonus pack), or one of the shotguns, but I've been using That Gun a fair bit since I bought it.

The Gobi Sniper Rifle. Looks awesome, kills fast. I like That Gun for its quirk.

Just found "Pushy" last night by accident, my character is only 30 melee and this thing still kicks (punches) ass like no ones business.
The stealth crit kills are complete lol material.
I never even considered melee weapons till now, this will always be on my person at all times.

The Golden Dong! Hands Down :bowdown:

Ratslayer, no doubt. Then again, my one-shot-one-kill stealth style makes the list of unique weapons I actually use pretty short. (It's a unique, much more powerful Varmint Rifle that you find equipped with all the mods - silencer, night scope and extended mags.)

With the proper ammo type, I can one-hit kill (while sneaking) practically anything... short of the big meanies (deathclaws, giant radscorpions, humans in power armor, Securitrons, etc).

Splattering supermutants and nightkin with the gore perk and annabelle the rocket launcher is a laugh

both guns with camo. and anything engraved (like maria). dont really have a favorite.

i would like the garand, but the sights are eww

I love Lucky.
The looks itself on an ultra-setting makes me jizz.

I do love Maria, La Longue Carabine and, of course, the Alien Blaster.

So good that there's so many differing responses. I quite like the sound of the Ratslayer and the Gobi Sniper.

100 % Gobi Sniper !!! Cliff on the rock and shot with scope is my favorite ! ))

This Machine aka the Garand, that thing chewed up and spat out any foe I had faced, from the lowliest raider to the horrifying Legendary Enemies after they had killed the everloving s*** out of me and my companions a few times before I said "To hell with it!" and used up my small supply of 308 ammo. :bolt:

I found some golden boxing gloves in the the Lucky 38 casino... obviously they don't do much damage but I still think their quite cool :celebration:

The Mysterious Magnum is my favorite love how it plays the mysterious stranger music

Ranger Sequoia and Trail Carbine is my absolute favorites. Those Ceasar guys wont even see it coming before I blow their heads off.

The chopper (a cleaver)

I managed to find it early on in the game and became my best weapon from level 5ish to level 10

Nothing like some good ole head decapitation "chop, Chop, CHOP!"

I like Lucky .357. Simple, easy to repair, plenty of ammo, great iron sights, quick reload and crits like crazy. I killed Deathclaws with it, Fiends, Cozadors, Super Mutants, Spore guys, Legionaries... everything. Having a high AP character I could give 6 crits to the head of whoever came close - and there aren't many things that can survive that in the game.

Mines Probably Maria, used even at level 26

The All-American.

So far ive found..

Ratslayer, used this for a very long time.
Lucky, didnt use this once.
Annabelle, didnt use this once.
Thump-Thump, didnt use this once.
Ranger Sequoia, which im about to use but im not 100% sure its unique.
Knock Knock, havnt used once because I dont use unarmed or melee.

So I guess my answer is the Ratslayer.

Well crap. The only weapons I have discovered so far is "That Gun" in Novac and the "Paladin's Light" or similiar sounding name. The Power fist that lets out EMP shockwaves. I pretty much let Veronica take care of everything now that she has that slotted. Of course, I am only like lvl 10-11. What can I say, I am slow and don't have lots of time!

i've been using the all american and AER14 prototype...so where does one happen to stumble upon the gobi sniper rifle?

I've probably used Lucky more than any other unique. Great gun.

Thump Thump and Love and Hate XD theres more I just dont know the names