Most of the time when I run across a dog, I'm able to stop and yell at them to go home or whatever. Usually, that's enough. But I've had a couple of notable interactions:
One time a dog came out after me on an empty road, from my front left, charging full tilt, barking. I wasn't going to be able to get by him, and I was going too fast to stop before I got to him, so I aimed my bike at him as braked yelling my lungs out at him. He ran like hell. (I don't recommend this technique, and I have no idea what came over me.) It was pretty funny. He was like "Dude! What the hell?! That's not supposed to happen!" I chuckled for a long time over it.
Another time, some little yappy dog came out after me and chased me pretty hard. I had no problems outrunning him, but he kept up for at least a quarter mile. I as I came around a blind corner, dog somewhere behind me chasing, a Range Rover was coming in the oncoming lane pretty fast...and I knew what was about to happen. So I slammed on my brakes and yelled at the guy driving "SLOW DOWN! there's a dog in the road around the corner!" and he just looks at me and keeps going at speed. He went around the corner and I heard his brakes squeal and the dog give a short horrible scream. It made me really sick and sad, because I'm a dog owner, and I love dogs, and that was someone's beloved pet. Totally the owner's fault, but still. Sometimes **** just happens, and maybe the dog got out of the yard this one time. I don't know that the dog died, but given how small it was, I would think so. I didn't go back and check.