Seen cats do it. Little twitches, especially the paws, and making noises. One cat I had made a specific noise it only made when hunting birds.

Absolutely. My Dog moved more and made more sound than the cats do - but the cats definitely do so too.

It's adorable and if I put a hand on them gently it seems to quiet them back to normal sleep.

All the time

I love it when I see dogs dreaming. Same thing -- running and barking.

Yes I have had a couple dogs that run and bark in their sleep, Not really running but paws twitching as if they are running, it's so cute. I have had to wake them up out of a REM nightmare when it appears my dog is having a bad time, crying and such. They are Terriers.

Her Royal Cuteness dreams all the time.
I've never seen a cat, or any other animal do it.
That's not to say they don't, but I've never seen it.

Dogs do that frequently. Had an old terrier dog that passed gas in his sleep, which was OK as long as he wasn't sleeping with me. One day we were on the sofa, I was watching TV, he was curled up behind my leg, sound asleep. He farted out loud, loud enough to startle himself. He acted like someone had fired a shot, as he jumped up from a dead-sleep and ran. I laughed and laughed. It was hilarious.

Had a good laugh at my Airedale dreaming just this morning, watching her 'talk' and chatter her teeth. Funny beastie.