Hands down the belt drive. I work at a garage door dealer as a tech and installer. We do not even carry screwdrives. The screw drive is quiet at first, but after a while it gets very noisy. Also a screw drive, as it gets older can sling the lubricant in it all over the face of your door. When you see a door with a streak down the middle, you can bet they have a screw drive. Also, they deliver more torque than the other two. So when your spring breaks, or something in the doorway (but not blocking the eyes), it is harder to stop which can cause more damage to your car, door, or whatever else is in the way.
Chain drives are very good openers. They are durable and last a long time. I have yet to see a broken chain except when it was installed in a constant wet environment, or it was a 30 plus year old commercial opener. That said, I the only chain opener i saw with a broken "chain" was a craftsman, chamberlain and Overhead brand opener where they use half cable and and half chain. Where the cable connects to the chain the cable snapped.
Our best seller and my personal favorite is the belt drive. It is much quieter and lasts just as long as a chain drive. The belt does not, or I have yet to see one "stretch". The Craftsman, chamberlain and Liftmaster ones have spring tensioners on them, so they always stay tight. As an added bonus the Chamberlain and Craftsman openers have 10 year warranties on their belts. The Liftmaster has lifetime warranty on the belts.
Like I said hands down belt drives are the best bang for the buck. Or if you have the money and your garage is wired properly check out the Liftmaster Jackshaft opener. I would stay away from the Wayne Dalton idrive. They are junk.
***Check out the belt drives with battery backup***