My personal opinion. Certainly, this is open to debate, since personal preference plays a big role in deciding what weapon is best for the individual.
AR: Harrier (great DPS, low weight), PPR (very useful in a variety of situations), and GPR (while the Harrier supercedes weapons like the Typhoon, Saber, Valkyrie, etc... in a variety of roles, the GPR still remains better in a variety of circumstances).
SR: Black Widow (makes great use of the TC damage bonuses, kills both mooks and bosses well), Indra (light, and strong), Kishock (bleed damage, can be fired from the hip with no penalty, does a ton of damage, doesn't weight too much, can be used on non-Infiltrators).
SMG: Hurricane (great DPS and low weight), Collector SMG (don't have it, but some people love it, apparently. If it's as horrible as some people say, replace this with the Hornet for being similar to the Hurricane, but slightly more controllable at the cost of DPS), and GPSMG (while this weapon has an awful damage output, it applies ammo powers very well, making it a good sidearm for certain characters. Locust can also be useful thanks to it's decent damage output with headshots, which are easily obtainable thanks to the low recoil).
Shotguns: Reegar Carbine (I hate RNG for not giving me this thing. Great DPS, especially against shields and barriers, while still being useful on a variety of classes due to it's low weight), Piranha (similar DPS to the Reegar, but with more range at the cost of no shield/barrier multiplier and not shooting electricity, which looks awesome), and the Claymore (does an incredible amount of damage with a single shot, allowing this weapon to be great on low-durability builds. Also, it's the Claymore).
(It's a bit criminal that I can only pick 3 shotguns. The GPS, Graal, and Disciple (with Shotgun Omni-Blade on a melee build) all are honorable mentions as well).
Pistols: Acolyte (the ultimate caster weapon. Completely drains shields on many enemies while also setting up a FE with Incendiary Ammo, meaning shoot -> ability is enough to kill a lot of enemies. Aside from that, using Barrel + ULM makes this weapon into an incredible sidearm which, should you not already be using this weapon as a primary, gives you an incredible way to deal with shields), Paladin (low weight, great damage output. The Talon does a similar job very well, but the longer range of the Paladin makes me lean towards this weapon instead), and the Scorpion (from what I understand, it's the best AOE weapon in the game. Great damage output against groups, low weight, ALWAYS procs ammo effects on a direct hit, etc... If you're not going to use the Acolyte as a secondary, this can be a great alternative).
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Hurt my soul a bit that the Talon, GPS, Javelin and Graal didn't make the list, but I stand by my choices. These really are the best weapons in ME3 MP.