It really depends on the mission as well as the class you're playing, though I do tend to use certain squadmates on certain missions regardless of what class I'm playing. I will say that I know what I'm talking about with ME2 as I've successfully completed Iron Man (no deaths) runs on Insanity with Sentinel, Soldier, and Engineer and come close with all 6 classes.
Jack is almost always my go-to for Lair of the Shadow Broker because her Warp Ammo is useful for getting through Tela Vasir's barrier quickly and you also fight multiple Vanguards with barriers in the mission. Her Shockwave is also very handy for knocking people into the air for instant kills in the Broker ship exterior segment.
Kasumi is basically a "don't leave home without her" choice for any mission vs. the Collectors because Flashbang completely disable Harbinger and makes him unable to hurt you.
Grunt is likewise a must have for the Derelict Reaper mission.
In general, always have at least ONE squadmate that has a disabling ability like Pull, Throw, Flashbang, etc. preferably on a short cooldown (conc. shot has a long cooldown of 12 seconds). The squadmates with fast cooldowns are also always powerful: Jack, Kasumi, and Samara have passive bonuses to cooldowns.
In the early game Zaeed and Garrus with Zaeed's disruptor ammo or one of Shepard's ammo powers kick some serious ass with Incisor snipers, as in the early game you also get access to several Sniper Damage upgrades. I usually take both of them along for Project Overlord and also Jack's recruitment mission.
I strongly recommend recruiting Grunt to take him along on Dossier: Archangel, he's incredibly valuable in the garage sequence due to his tankiness. The other squadmate I take is usually either Jacob or Zaeed.
Really there's more to tell than I possibly could but if you have specific questions I can answer.
