Varric is probably his closest friend of the Inquisition.They were in it from the beginning, both with an earnest, if wry, perspective on the entire thing and somewhat united in their views on the Chantry. They have an easy camaraderie, and Solas respects Varric’s seemingly unending pool of empathy. Though they do not always agree, I would argue that in a way they are sort of the team parents.
This is best illustrated in their relationship with Cole. Solas is a surrogate father to him in many ways. More reserved in his emotions than Varric, he still very clearly cares for Cole and will defend him at all costs. Cole clearly respects his opinions and sees him as something of an authority on magic, spirits, etc. They understand each other, and I believe Solas makes Cole feel more comfortable. Solas is an intermediary between both worlds, and to a fish out of water like Cole this is comforting. They speak the same language.
He greatly respects Cassandra. He finds many of her views misguided or a little naive, but he recognizes in her a genuine search for the truth, and admires that she will change her mind if she learns she is wrong. They both hold each other at a respectful arms length, given their natures, but there is nothing cold between them.
Solas gets along very well with Blackwall and Cullen. They are soldiers, hardened, practical men in whom he sees a great deal of himself. He talks to them as equals, sharing that ‘band of brothers’ bond forged in battle. Their talk is what you might hear in any soldier’s camp: ‘gentle’ ribbing, comments with nearly no context, wry observations, and a shared, oft unspoken tragedy. Particularly with Blackwall, which is why his ‘reveal’ hits Solas so strongly. He had seen Blackwall as a better Solas, and this revealed they were very much the same.
In a way, the same goes with the Iron Bull. Solas has no respect, and outright despises, the Qun, but he does see the Iron Bull as a worthy man, and comes to respect him as well. In a way, Solas’ own lumping of everyone under the Qun effected his perception of the Iron Bull, until he bothered to see him outside of it. He recognizes in him a great intellect underneath his boisterous exterior, and a greater pain. They may not be the warmest companions, but there is little tension when they are in the party together.
Unlike Solas and Sera. She is very difficult to get along with, and very stubborn, which is not helped as Solas is also very stubborn and convinced he is right. He is constantly trying to ‘right’ her, bring her to an understanding of her people, their innate magic, and what he sees as her potential. Regardless of whether she wants it or not. It is also a test of his patience, for her attitude is not an easy thing for him to swallow.
Solas and Vivienne are ever at odds with each other. They are both mature adults who know what they want and what they believe to be entirely true. While he respects her intelligence, he sees her ambition as extremely dangerous (probably in no small part because it was like his when he was younger). I believe she reminds him of the old gods, powerful mages lording over others under the guise of ‘order’.
Which leaves Dorian. Their relationship is, on the surface, highly antagonistic. They are always trading barbs, and despite all of Dorian’s attempts at ‘reaching across the aisle’, so to speak, Solas always returns with a snide, harsh comment. Underneath that, though, there is a mutual admiration and honest affection. Their constant arguing - over magical techniques, history, everything - is how they’ve grown as friends. They may not be warm to each other, but anyone who mistakes their bickering as anything other than friendly is mistaken.
His relationships with Leliana and Josephine are much harder to analyze, as beyond the codex entries from Leliana’s perspective there is very little interaction between them. I think, going off of nothing, that he is probably wary of Leliana. If anyone can sniff him out, it’s her. He doesn’t want to give her any reason to poke into the stories he’s told of himself. He is courteous, respectful, and entirely professional.
UPDATE - theharellan said that: “Josephine says she likes his stories and finds him very wise and interesting when you ask her about him.” I like to imagine him wandering outside of his rotunda more often than might seem. He loves being around people, and he loves talking and listening and Josephine is such a champ that I can see them passing each other in the hall, pausing to say hello, and a half hour later they’re still chatting.