I personally think Sona is excellent for Dom. Aria of Perseverence, while not an amazing heal thanks to Dom's heal scaling, gives a large armor/MR buff for a few seconds to both her and her lucky teammate. Her ult wins teamfights, period, and Q > powerchord Q does terrifying amounts of damage. This is in addition to her "weak" movespeed buff, which she grants to her whole team and lets them travel the map together better and faster.
She doesn't win games on her own, but she makes a good team great. Pair her with a strong buddy to support like XZ, Trynd, Riven, Vayne, etc. and she will give her buddy a lot of assists.
I personally think she's second-string, but she's totally usable. Post-level 11 her range is ridiculous, and her rocket jump lets her get into good positions or escape bad situations. I think Vayne and Cait are better Dominion picks, but she's not bad.
Characters that I think don't do well are much in the same vein as the previous list:
Amumu - hard to play well but totally viable
Annie - not enough frequency of bursts; once she fires she's spent for a long time, making her very baitable
Cho - kind of a bad pick except in a few specific matchups where he is very strong
Eve - not terrible, but takes a lot of work to do well; most eve players think that stealth means backdoor all game
Fiddle - smaller-scale teamfights and the general prevalence of CC make him easily countered; still strong if not countered and multi-target silence is always amazing in teamfights
Garen - much harder to decide when to commit to a fight in Dom for him
Katarina - extremely reliant on ult to do damage; ult easily countered
Morde - doesn't play well with others top; has many counters bottom, scales badly
Nasus - much like Cho, only good in a few specific matchups
Nunu - see Kat; bad kit for Dominion
Shyvana - never gets to the immense lategame power that she can reach in SR; in rare situations, she can be absurd
Teemo - can be played well, but the signal to noise ratio on him is extremely high
Vlad - playable, but much harder to win with than other champs; his ult still really wins fights
Naturally, some people disagree (I think Fiddle is weaker than popular opinion, I think Sona is underrated, etc) but I think that the available pool of champs is pretty large.