My dogs do a lot of weird things, but the one off the top of my head right now is something the girl in my avatar does several times a day. I can't describe it very well, but she will stretch out her front paws and lean back with her butt high in the air (like a downward facing dog!) and then a few seconds later reverses position to where her head is high in the air and her back legs are stretched out way behind her. Then she'll plop down on the ground really hard and sigh in contentment.
I've obviously seen dogs stretch before, but I've never had one that did it so precisely in these positions on a regular basis like her.
Our dog Dash does the same thing, but only once a day as soon as he gets up and comes out of his crate each morning. We call it sun dog rising

and then he lays flat on his belly with his legs straight out in front of him and pulls himself along, dragging his back legs, we call this his paddle. Next he lays on his side and curls up and scrunches around on his side, first one side and then the other, we call this the boof. Every morning, the same routine, and not again for the rest of the day.
He is the only dog I have ever met that does not respond in any way to the kissy sound made to normally call a dog. He stares at you like you are insane, but he won't take one step towards you. I really believe he is a cat in a dog body.
He will go to the back door and whine to go outside, but when you get up to take him out he runs and grabs his KONG and drops it at your feet to put a treat in it. Smart dog!
He also drops his KONGs from the top of the stairs so they bounce all the way down and roll on the floor, just to get our attention. If we ignore him, he will go down and get it and then drop it from the top again, and again. We call them his grenades. He has about 4 different shapes and sizes of them and pretty much daily we find one in one of our shoes. I've never seen him put them there, so I don't know if he drops them in the shoe or if it bounces and lands there, but it cracks me up that he does it.
My DD used to ride the bus and every day at 4pm, he went to the front door to go out and meet her. It took him probably a year to break that habit when she didn't ride the bus anymore.
He loves broccoli and carrots, he will come from anywhere in the house when he hears me take either from the fridge. I have tried sneaking it as quietly as I can, it doesn't matter, he will hear it!
Every night he goes to the kitchen between 8 and 9pm, and if no one has made popcorn he whines until someone does. I finally have started keeping a bag of air popped ready, so if no one has it he can at least have a small treat. We can't call it popcorn anymore, it is sodawheat.
If we go somewhere and pet another dog, we get a total sniff down when we get back and he is usually miffed for a bit with us. I always tell him I cheated on him, he doesn't like that

He also knows if we went to the movies and had popcorn without him, he really gets miffed then

I've started bringing him home leftovers in the bag.