I was a Mage, basically a Spirit Healer. By this I mean I only needed a Rogue for Lockpicking and a Tank Warrior, though I liked to have a second Mage oftentimes just in case.
I tried to use everyone, but it was really hard to stick with this ideal.
Alistair - Yes. I used him a lot. He was my Warden's friend and unrequited crush. Bwahaha. Tank. In party most of the time, leaving only 2 other slots as variables. Sometimes replaced by Oghren, Sten, or Shale if I thought I could handle that.
Morrigan - Yes mostly. I used her a lot, but didn't feel like I needed to. Friends with her but she's not hinged on my Warden's story. I could just as well bring Wynne instead, or even no Mage.
Dog - Almost never used. Sometimes I tried. Maybe I could bring him a lot on a story alt, someday. A story alt that loves dogs or something.
Leliana - Used early on. I needed a Rogue, I liked her well enough, but I dropped her once I could level Zevran's lockpicking. Nothing against her.. she's just not my Warden's LI.
Sten - While I didn't mind him, I also couldn't relate. He wasn't a big part of this story. But good relations and I took him out once in a while.
Zevran - Definitely. While I didn't use him ALL the time, he was my Warden's LI and a needed Rogue (once with Lockpicking) so it stayed. Kept him once I knew I could open chests with him, so I'd count him as a near permanent one.
Wynne - I wanted to use her more, but this was limited because I already had one Spirit Healer. I don't dislike her as much as many others. In fact, I like her. Just wasn't as much in the cards for this playthrough.
Oghren - Well I didn't hate him and I liked having an axe wielding dwarf on the team so he was around sometimes.
Shale - Loved her! Can't say she was a fixture on my team, but she's on the upper end. Very fun, and acted as secondary tank-ish when needed.
Loghain - Killed by me.
TLDR:
Always Used - Alistair, Zevran
Sometimes Used - Morrigan, Leliana, Sten, Wynne, Oghren, Shale
Never Used - Dog, Loghain
Though 'Always' and 'Never' are not actually absolutes, and 'Sometimes' varies a lot.
A common team: Alistair, Zevran, Shale
This changed sometimes when I wanted a human female present or just variety. I only really shied away from Dog (and maybe somewhat Sten and Oghren as I already had my clear Warrior in Alistair; I considered Shale unique).