I'll go by each category, since I can't choose just one. I'm going by what I do well with in Hardcore mode, so these may not work for everyone in every mode.
AR: SWAT, definitely. Select Fire is ridiculous on this gun, and I can spray a whole clip at SMG speeds if I have to. It's my deadliest weapon, by far. You want perks that decrease anything that can affect your accuracy, and attachments that improve it. Toughness and Tac Mask reduce anything that can throw your shots off, and Cold Blooded allows you to run around the map and get to a good postion. Quickdraw Handle lets you acquire targets faster, and a tactical grenade or two can stun opponents for long enough to get the kill without taking damage.
SMG: Vector K10. With Quickdraw Handle and Long Barrel, I can hold down whole sides of small maps like Slums with ease. Use any perks that help you get around the maps fast and quiet to surprise your opponents, like Lightweight and Extreme Conditioning. Fast hands helps you throw your grenades faster into rooms or over walls to get your enemies weak and defenseless for the quick kill shot.
LMG: QBB LSW. It's a bullet hose, and with Quickdraw and Stock, it can be as deadly up close as it is from afar. Any perks that allow for faster movement are a lifesaver, and allow it to compete with ARs in the mobility department.
Sniper Rifles: The SVU-AS is great for me because I have the trigger finger for fast follow up shots, but the XPR-50 is my second favorite because of the larger OHK area and ability for second shots that are almost as fast as the SVU. Like with the SWAT, any perks that decrease anything your opponents can do to throw off your accuracy are good. Toughness and Tac Mask are good, but more useful in Core mode where you can take more shots before you're killed. Here, I use Cold Blooded to peak around/over objects and not have my position given away by a red name over me.
Shotguns: I don't use these very much, but the S12 with Long Barrel, Fast Mag and Laser Sight is a solid CQB option. Again, movement perks are perfect for this weapon, for getting in close and decimating your opponents.
Secondaries: Including all of the options at once, the pistols are by far the best and most versatile options. The Five-Seven and Tac .45 are both solid choices, but I love the Executioner on my sniper classes with Fast Mag and Laser Sight for a close quarters surprise. The launchers are only useful for taking out killstreaks, as some people run Flak Jacket and will take more than one rocket to kill.
Specials: The specials (Crossbow, Ballistic Knife and Assault Shield) are only good for trolling, really, so give them a pass.
Weapons are quite balanced. Though on certain maps the Remington shotgun outclasses almost any gun. Hijacked, Nuketown and the smaller maps with tight corners in general. Smgs in general are badass and put people on the floor quickly. Assault rifles aren't bad by any means... Just need to know how to play with them.
I'd like to say the Skorpion, but it just burns through ammo. I'll put my vote for the FAL. Yeah, it's effectiveness in CQB is limited, but you won't get caught reloading, and the damage can help you in a pinch. Put on a target finder and stock and perhaps quickdraw and you'll never get caught off guard.
Other than that I guess I would say MP7. It's a solid, well rounded weapon for almost any situation.