Updated 16 October 2022 for the season 1 changes. Changed the old attributes to grayscale, and added the new effects from the new changes.
Some changes are only for the text. Most mystic runes are now solo, meaning, only one axie can use that in a team, but other axies can use the other mystics. This mechanic was already present in the beta, but only for the dawn class mystic. Aqua runes: turned out the high extra hp was too strong for aquas, so it was removed from all the runes, also the bubble was turned into offensive mechanic against high hp and doesn't provide defense anymore. Several aqua cards were nerfed from good utility into bad deck thinning cards. For example tiny fan was a build defining card, now it gives a single +4 attack bonus one time. Beast runes got a small nerf, and some rage mechanics nerfed. Pangolin still seems to be the strongest rune in the season 1 meta. The top 25 players are mostly 3x beast users, 2 with pangolin. The bird 'venom born' rune turned into the 'incubator', giving extra hp for summons, making them last longer and survive some other dangers like aoe or curses. 2 other bird runes got a buff too. Plants got the old aqua rune effect and the mystic effect became the epic rune.
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Each axie can have a rune in v3, there is a locked slot so probably two later on, there is no way to unlock those yet. Runes are buffs that give permanent stats or semi-permanent buffs to axie or all axies even affect opponents. There is no requirement of purity as long as you own these runes, any axie can equip any rune of it's class or any neutral rune, but only one axie at a time. The card "Oranda" could actually remove some of these buffs, this was fixed in the patches, now these runes last all through the combat. Bumpy, sakura (powers) can't be removed either. Higher rarity runes are usually stronger, mystics can be used by only one axie of that class.
There was a bug (back in march) allowing people to click the moonshards in inventory then crafting runes and charms took away moonshards in the bonus slots, and crafting 1 cheaper item or disenchanting one rune gave back almost the full cost. On top of that it was possible to make more than one at a time 150-200 sometimes froze the client but was no side effect to it. A lot of people realized it and made the best runes and charms already. Now the new update didn't take away the extra moonshards or runes created this way they even added 20 of each rune for everyone to try out. The runes expired with season one so you got to grind up the levels and craft them again. Runes and charms can be sold on the marketplace (unless it's common or non seasonal), but you got to wait 7 days after received or bought, to sell it back. Also they have to be rare or higher by the looks of it. Most runes will expire at the end of the season and all you can do is disenchant them. So it will not be much value to buy them at the end of the season.