I think sleeping on their backs has to be a breed favorite. Max use to sleep on his back all the time but since he developed neck problems he doesn't. Now Katie...always on her back, even in her crate at night she is 4 paws up.
Ellie isn't so much one to sleep on her back, but she will make herself as small as possible. To elaborate, I bought a doggy bed a couple of years ago with the intention of letting both of the dogs share it. It's large enough comfortably fit a full grown Labrador. Does Ellie stretch herself out and lounge in style? Nope! She curls up as much as she can and sleeps only in the far corner of it! XD Heck, she doesn't even let Yuki share it with her. She hogs it all just for that one tiny little corner.
When she's downstairs, she usually just flops down onto her side or lays her head down on her paws to snooze. Well, unless she decides to use her crate. Then she'll shove herself into the farthest corner again, hehe. Sadly, she hasn't slept on her back since her puppy days.
Rosie sleeps crossways in the back of her crate (we leave the door off for her) flat on her back with her head at a 90 degree angle to her body. Rocky sleeps right side up with his head going one way and his front feet going the other. He's slept this way since he was a baby and he's pushing 7 years now.
Wally doesn't sleep on his back but rather belly down on hard wood or ceramic. He loves the coolness.
He usually curl up on his bed with his tail close to his nose, or snuggle in a corner of the living room behind the sofa, on his side. But occasionally he loves to have his head resting on his bed and the body on the ground.
Denis....Max does the opposite. He will sleep with his whole body on his bed and his head on the floor. Katie likes to squeeze between the bed and the TV stand and hide her head under the bed.
Most of mine sleep in the open kennel and many times on their backs. Wynn at 9 still sleeps in his puppy bed with his head hanging out. He has big beds but prefers his baby bed:)
When Dodger was a puppy he use to sleep on his back all the time or he'd create a circular nest/bundle and sleep in a curled up ball. As he got older, he sleeps about 70% on his back and the other 30% in the most weirdest and hilarious position he can think of. My husband once caught him with his face planted against his crate door and his romp in the air. It was absolutely priceless!
Sometimes Ruby and Cassie sleep on their backs, but not often. Usually it's if they roll over in their sleep.
Cassie had already staked out a corner of the tiny, tiny back bathroom as her Nest. Ruby approached with trepidation and when she finally weaseled her way into the den back there, she took up a position in behind the toilet. She shelters between the edge of the shower and the base of the toilet, which is about eight inches wide. Guess she figures this makes it harder for Cassie to remove her head while she's sleeping. ;-)