Leashes are for dogs... Ok, kidding but really, for surf, small waves, worse thing ever. If you are learning to surf and its two foot mush, you are falling off in 2 feet of water, you really want to be attached to a sup? It isn't going far, you ain't gonna drown. But that leash will jerk that board right at your body after you fall and its coming back to you tail first with that nice pointy fin with its sharp edge...I have seen so many injuries from this very thing.
"But there are other surfers/people in the water around!"
If that is the case, you shouldn't be out there as a beginner. Go down the beach and find your own little wave. Seriously, this is what has caused the hate for sup's at some spots, beginners who paddle out in a crowd and running into everything and every one. I have schooled many surfers and paddlers on this issue, nicely of course.
Heavy swims and big waves, heavy rocks/cliff faces on the beach, sure, straight leash at a length of 2' over the board length, ankle only (You do not want a calf leash in surf over 6', this is how you dislocate knee's) 5/8ths thick if you can find it. If you are going to surf over head plus waves, double leash plugs are good, a 10' by 30" surf sup will drag you, a single plug will pull out and 3/8ths or 10mm leashes will snap.
Garret had a video up showing him hanging on to his leash underwater while the board was in the whitewater doing its thing. He was using the paddle blade in his hand to steer. I can't find the video he posted but I personally have eaten shit in 15' plus surf, you will think hard about hitting the quick release. I was pulled along for a good 60 yards before I pulled the cord and came up for air.
For slow rivers, I wouldn't bother. Bays, nope, unless the wind is blowing. Open ocean, yes if the wind is blowing. Coiled is the ideal for any flatwater paddling.
I have seen an inflatable doing 15 knots going downwind without a paddler, its funny until it isn't and the guy is waving his arms for help. But even hard boards will take off in the wind like they have a sail up. I have done some offshore paddling, at times 12 miles offshore and always with a leash because of this.