Of course the best to start with Anne McCaffrey.
So good choice there. No one's mentioned this one: Elric of Melnibone has dragons in it.
Also
Voices of Dragons by Carrie Vaughn's about a girl who lives in a world where dragons have taken over Canada. So, check that one out? I think I recall hearing she was working on a sequel to that one.
There was one I read ...shew, a long time ago called
The Glass Dragon/The Dragon Nimbus by Irene Radford.
Then the Dragon Prince series by Melanie Rawn comes to mind. Those first three books are on
Kindle. The second series, Dragon Star doesn't seem to be.
And last but not least I just finished a fantasy romance for review--Marsha A. Moore's
Seeking A Scribe has some great (little) dragons in it that I just adore. I'll have the review on my blog sooner or later. I just gotta get off my behind and do it.
Anyway, I hope you find something here you like.