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.