• Avenger
  
• Vindicator
  
• Mattock (Weapon pack DLC)
  
• Collector Rifle (Deluxe Edition)
  
• Geth pulse rifle (Hardcore or Insanity)
  
• Revenant (Shepard Soldier only)

Used by : Archangel (early), Zaeed (early, DLC), Grunt (early), Samara (late), Legion (late).

They’ll make good use of each, I guess. But Avenger is ♥♥♥♥. That leaves the four others. The best is probably the Mattock, if you have it. The Vindicator is also a very good choice.

If your Shepard uses an assault rifle : the Vindicator and the Mattock the best. You can also, as a soldier, get the Revenant LMG on the Collector ship. The Geth Pulse Rifle is a little bit more effective against shield and barriers, but nothing crazy.

With other classes than the Soldier, I wouldn’t recommend taking the Assault Rifle training as a specialization, since you will already have the SMG to do the same things.

• Mantis
  
• Viper
  
• Widow (Shepard Infiltrator/Soldier and Legion only)
  
• Incisor (DLC)

Used by : Archangel (early), Zaeed (early, DLC), Thane (late), Legion (late).

All your squadmates have sorts of aimbot and autoshoot, so they’ll be deadly with sniper rifles. The Viper will not be that effective on them, but the Mantis (and the Widow for Legion) will.
The best choice is the Incisor : squadmates don’t have recoil, so they will do a triple headshot with that weapon, something you will never be able to do with Shepard. Also, they have infinite ammo.

If your Shepard uses a sniper : the best will be to use the Viper, so you can use the slowmotion (due to the Infiltrator’s passive OR the Soldier’s Adrenaline Rush) to do multiple headshots. Mantis and Widow are also a good choice, if you don’t like the Viper.

With other classes, I wouldn’t recommend taking the Sniper Rifle training as a specialization, since it’s basically never interesting (and shotgun is better).

• Predator
  
• Carnifex
  
• Phalanx (Weapon pack DLC)

Used by : Miranda (early), Jacob (early), Jack (early), Mordin (early), Kasumi (early, DLC), Tali (late).

Basically, the Predator sucks, and your squad mates won’t benefit from the Phalanx’s accuracy bonus (laser targeting), since they don’t use it. So give them the Carnifex, it has a higher DPS.

If your Shepard uses a heavy pistol : the best choice is the Phalanx : really deadly, and you have the laser visor to pinpoint everything. If you don’t have it, the Carnifex is still great.

• Katana
  
• Scimitar
  
• Geth plasma Shotgun (Weapon pack DLC)
  
• Eviscerator (Cerberus Network DLC)
  
• Claymore (Shepard Soldier/Vanguard and Grunt only)

Used by : Grunt (early), Jacob (early), Jack (early), Tali (late).

Basically, your squadmates don’t know how to use shotguns. They'll try to use ‘em from a way too great distance. Which means they will be efficient only with the most precise shotguns : the Geth Plasma Shotgun and the Eviscerator. If you have neither, well, equip their other weapon (Assault rifle or pistol). Do not give the Claymore to Grunt, as he won't know how to use it.

If your Shepard uses a shotgun : he’ll do best with the Eviscerator or the Geth Plasma Shotgun too, since they deal more damage than the Katana or the Scimitar. And they are more accurate as well.

If you can unlock the Claymore : with the Vanguard, take it. Nobrainer choice. If you’re a Soldier, well, the Claymore isn’t that useful compared to the Revenant or even the Widow. But you can still take it if you want to.

With other classes, Shotgun specialization is probably the best choice, since this kind of weapon is rather unique in the game. The Adept, Infiltrator, Engineer and Sentinel can really do wonders with a shotgun in their hands. That gives them a little bit more freedom in their gameplay too. And they don't need an assault rifle nor a sniper rifle, since they already have the SMG and the heavy pistol.

• Shuriken
  
• Tempest
  
• Locust (Kasumi DLC)

Used by : Miranda (early), Mordin (early), Kasumi (early, DLC), Samara (late), Thane (late).

Your squad mates don’t use the Locust correctly (they just fire small bursts, instead of suppressive fire), so DO NOT give it to them. Give them the Shuriken until you unlock the Tempest, which is the best for them. They won’t have any recoil, so they’ll do wonders with this gun.

If your Shepard uses a SMG : the best choice is to get the Locust ASAP. It’s the best SMG : absolutely deadly and extremely accurate as well.
If you don't have it - or if you don't want to use it - the other two, the Shuriken and the Tempest, are great weapons as well. For close combat classes, like the Vanguard and the Sentinel, they can easily replace a Locust.

• Grenade launcher
  
• Rocket launcher
  
• Avalanche
  
• Cain
  
• Collector Beam
  
• Singularity Projector (Pre-purchase DLC) - Not tested
  
• Arc Projector (Cerberus Network DLC)
  
• Flamethrower (Zaeed DLC)

In high difficulty, the best is to use the Avalanche. It has a great crowd control since it can freeze enemies (for a very long time), and it has a great damage multiplier against defenses (shields, barriers, armors) as well. Also, any frozen enemy will take +100% from “powerful” attacks (Concussive shot, Throw… funny on the Adept!).

The Arc Projector is a fantastic crowd-control weapon, and it does wonders against synthetics and shields. It’s like ME2’s Arc pistol, for those who played ME3.
The Collector Beam is also a great choice to deal with heavy shields and barriers. Extremely precise, which can be useful if you haven’t got the Locust or an assault rifle (yet).
The starting grenade launcher is OK.
The missile launcher and the flamethrower aren’t interesting.
The Cain won’t be THAT effective, because you’ll never have enough ammo to use it easily. However, you might wanna take it to deal with Praetorians…

That's it. Hope this helped !