I don't know about you guys, but I like to kind of roleplay the weapons that my squad mates use. I usually cycle through the weapons for each of them, so that they never take the same weapon on two missions in a row. It gets so boring when everyone's using the same weapon over and over again.
For Garrus, I like to give him the Phaeston Assault rifle and the Krysae Sniper rifle. I choose these because they're turian weapons, and well, Garrus is a turian. Sometimes I'll load him out with a Vindicator and an Incisor, because those were the weapons I tended to give him in ME2. Or when we're dealing with Cerberus, I'll give him the Harrier and the Raptor. Since the files in LOTSB showed that he liked to kill killers with their own methods, I thought that Garrus would get a real kick out of taking out Cerberus troops with Cerberus guns.

Same idea for the Geth Pulse Rifle and the Javelin when I take him to fight geth.
For EDI, I prefer to give her the Talon pistol and the Hornet SMG. EDI is, after all, an AI made by Cerberus, so I feel it's only fitting that she'd use Cerberus weapons. Or sometimes I give her the Carnifex and the Tempest, to emulate the configuration I usually gave pisol/SMG squaddies in ME2. When against Reapers, I give her the Executioner pistol and the Punisher SMG, because she can dish out some MASSIVE damage with those two weapons.
I usually let Liara borrow the N7 Hurricane SMG and the Eagle Pistol. Liara's always been a good friend to Shepard, and so being one of Shepard's best friends has its perks.

Alternatively, I'll give her the Phalanx pistol and the Locust SMG, because those were the weapons she always used in Lair of the Shadow Broker. Other times, I'll just give her the Blood Pack weapon duo and call it a day.
For Ashley, I always interpreted Ashley as a lady who LOVED the big guns. So I usually give her the Revenant and the Widow for all situations. Or, since she's an Alliance soldier through and through, I'll give her the Saber and the Black Widow, two Alliance-made weapons that do some massive damage. She hates Cerberus, though, so she never touches the Harrier or the Raptor, but she never passes up an opportunity to use the Geth Pulser Rifle and the Javelin. Or sometimes she just gets the Avenger and Mantis, the most iconic weapons in their class respectively. When she regains Shepard's trust, Shepard lets her borrow the N7 Typhoon and the N7 Valiant.
As for Kaidan, I always interpreted him as being more precise, more accurate, and more efficient. So I usually give him either the semi-auto or burst fire assault rifles, and a semi-automatic pistol. He usually totes a Vindicator and a Carnifex, with a max level Ammo capacity upgrade to emulate the good old Spectre master weapons. Or, I give him the Mattock and the Phalanx, to make up for the fact that he missed out on all the sweet weapons in ME2. After regaining Shepard's trust, he carries the N7 Valkyrie and the N7 Eagle pistol. Other times, he has the Saber and the Paladin.
For Tali, I prefer to match the shotgun and the heavy pistol she carries, so that they have the same basic aesthetic. This means that she might be carrying a Disciple and an Acolyte one mission, a Venom and a Scorpion the next, and a Reegar carbine and an Arc Pistol the mission after. In critical missions, she takes the Geth Plasma Shotgun and the Phalanx pistol, her go-to weapons in Mass Effect 2. For Cerberus, she takes a Talon and an Eviscerator, and for Reapers, she takes a Wraith and a Paladin.
James is a heavy hitter, and he too loves the BIG GUNS. He'll take a Revenant and a Claymore and just tear through everything. Or I give him the Falcon and the Venom and watch the fireworks. Once he becomes an N7, I alternate between giving him the Pirahna shotgun and the Typhoon LMG for bullet raining or the Valkyrie and the Crusader for precision.
Finally, Javik almost always takes his Particle Rifle with him on missions, along with a Predator for backup. I'll also give him a Scorpion pistol and Falcon assault rifle for fireworks, or a Saber and an Executioner for heavy hitting. If the weapons from the Retailation pack ever come into single player, I'll give him the Collector Assault Rifle whenever he gets on Shepard's bad side.
So yeah, sorry for the long post, but I thought I'd share what I do with the weapons.