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What you see below is a grid of theoretically good ring sizes. I haven't tested every single one, but my calculator thinks they'll all be quite nice. As I find the time to weave sample segments, I'll post pictures of them.

I've found that weaves group up into "families" of weaves that all do well with the same size rings. So instead of listing the weaves alphabetically, I've decided to group them by ring type from thickest to thinnest, making it as easy as possible to figure out what to do with the rings you have. There are lots and lots of weaves that I don't have listed here. If you want something in particular, please let me know.

You can skip down to the millimeter sizes if you prefer that measurement system, but there aren't yet any sample pictures for millimeter rings.
 
Inch Sizes
I've been making very good progress on sample pictures for many weaves in inch sizes. I will someday complete this set, but by then there will be more weaves to test. A chainmailler's work is never done!
  21ga 20ga 19ga 18ga 17ga 16ga 15ga 14ga 10ga
jens pind   6:20 7:19 8:18 9:17 10:16 11:15 12:14 19:10
birdcage
euro 4-in-1
turkish round
6:21 7:20 8:19 9:18 10:17 11:16 13:15 14:14 22:10
half persian 3-in-1
snake
7:21 8:20 9:19 10:18 11:17 13:16 14:15 16:14 26:10
box
euro 6-in-1
half persian 4-in-1
oval
8:21 9:20 11:19 12:18 14:17 16:16 * 18:15 20:14 ** 32:10
HP 5-in-1 sheet
full persian
10:21 11:20 12:19 14:18 16:17 18:16 20:15 22:14 35:10
* I like 15:16 better than 16:16 for half persian.
** I like 19:14 better than 20:14 for half persian.


MULTI-SIZE WEAVES:
  21ga 20ga 19ga 18ga 17ga 16ga 15ga 14ga 10ga
parallel 6:21
10:21
7:20
12:20
8:19
13:19
9:18
15:18
10:17
16:17
11:16
18:16
13:15
20:15
14:14
23:14
22:10
36:10
dragonscale   8:20
12:20
9:19
13:19
10:18
15:18
11:17
17:17
12:16
19:16
14:15
21:15
16:14
24:14
25:10
38:10


MULTI-GAUGE WEAVES:
  21/19ga 20/18ga 19/17ga 18/16ga 17/15ga 16/14ga
japanese 12-in-2 6:21
10:19
7:20
11:18
8:19
12:17
9:18
14:16
10:17
15:15
11:16
17:14

   
Millimeter Sizes
Danger, danger, danger!! I've only tested a couple of these sizes with pliers. The vast majority of these recommendations are based only on what my calculator tells me will work. That said, my calculator is usually pretty good with such things - so you'll probably be safe.
  21ga 20ga 19ga 18ga 17ga 16ga 15ga 14ga 10ga
jens pind   2.5M20 2.75M19 3M18 3.5M17 3.75M16 4.25M15 4.75M14 8M10
birdcage
euro 4-in-1
turkish round
2.5M21 2.75M20 3M19 3.5M18 3.75M17 4.25M16 4.75M15 5.5M14 9M10
half persian 3-in-1
snake
2.75M21 3.25M20 3.5M19 4M18 4.5M17 5M16 5.75M15 6.25M14 10M10
box
euro 6-in-1
half persian 4-in-1
oval
3.5M21 3.75M20 4.25M19 4.75M18 5.5M17 6.25M16 6.75M15 7.5M14 12.5M10
HP 5-in-1 sheet
full persian
4M21 4.5M20 5M19 5.5M18 6.25M17 7M16 8M15 9M14