Evolution tree

Every shape a delivered blade can take, by grade. Read left to right: the form shown is set by the highest grade the sword has ever reached.

Every delivered blade
Worn Sword+0Worn SwordRefined Sword+2Refined SwordDecent Iron Sword+4Decent Iron Sword

From +6 the lane sets the shape, and at +10 the hidden branch picks one of its two families:

Citadel Swords
Thick Iron Sword+6Thick Iron SwordPortable Iron Door+8Portable Iron DoorSomething's Off with This Greatsword+10Something's Off with This GreatswordDouble-Plated Greatsword+12Double-Plated GreatswordA Gate Torn from Its Hinges+14A Gate Torn from Its HingesFortress with a Handle+16Fortress with a HandleMountain-Crushing Citadel+18Mountain-Crushing CitadelContinent-Cleaving Iron Citadel+20Continent-Cleaving Iron Citadel
Holy Swords
Thick Iron Sword+6Thick Iron SwordPortable Iron Door+8Portable Iron DoorLightbound Silver Sword+10Lightbound Silver SwordBlessed Blade+12Blessed BladeDawnblade+14DawnbladeBlade of the Worthy King+16Blade of the Worthy KingSword of the Promised Dawn+18Sword of the Promised DawnExcalibur+20Excalibur
Sawblades
Angry Hunk of Metal+6Angry Hunk of MetalBiting Blade+8Biting BladeIs This a Saw or a Sword?+10Is This a Saw or a Sword?Plate-Chewing Sawblade+12Plate-Chewing SawbladeInterlocking Steel Sawblade+14Interlocking Steel SawbladeSteel-Grinding Shredder+16Steel-Grinding ShredderMountain-Cutting Saw+18Mountain-Cutting SawUnstoppable Gears of Ruin+20Unstoppable Gears of Ruin
Dragonfangs
Angry Hunk of Metal+6Angry Hunk of MetalBiting Blade+8Biting BladeYoung Dragon's Fang+10Young Dragon's FangDragonscale Blade+12Dragonscale BladeBristling Dragonblade+14Bristling DragonbladeWinged Wyvernblade+16Winged WyvernbladeBlade of the Dragon King's Breath+18Blade of the Dragon King's BreathDemon Blade: Dragonslayer+20Demon Blade: Dragonslayer
Runeblades
Shard of Magic+6Shard of MagicAwakening Blade+8Awakening BladeRunescribed Blade+10Runescribed BladeAwakening Runeblade+12Awakening RunebladeOvercharged Blade+14Overcharged BladeArchmage's Arcane Engine+16Archmage's Arcane EngineReality-Cracking Blade+18Reality-Cracking BladeAkasha, Blade of Genesis+20Akasha, Blade of Genesis
Splitblades
Shard of Magic+6Shard of MagicAwakening Blade+8Awakening BladeBroken but Still Together+10Broken but Still TogetherBlade That Hates Being Alone+12Blade That Hates Being AloneBlade That Commands Blades+14Blade That Commands BladesShameless Blade+16Shameless BladeLord of a Thousand Swords+18Lord of a Thousand SwordsSovereign of Blades+20Sovereign of Blades
+0 to +5

Every delivered blade wears the same three forms: Worn Sword, Refined Sword at +2 and Decent Iron Sword at +4.

+6, the lane

The shape at +6 comes from the delivery lane. Each day's swords are dealt to the three lanes in a shuffled order, so a delivery of three always has one of each.

+10, the family

Each sword also gets a hidden 50/50 branch on delivery. At +10 the lane and the branch together pick one of the six families, and from there it changes again at +12, +14, +16, +18 and +20.

Merchant lines

bought from the ghost merchant, new form every 2 grades
World Tree swords
Fallen Branch+0Fallen BranchSurprisingly Sturdy Branch+2Surprisingly Sturdy BranchClub That Refuses to Break+4Club That Refuses to BreakLate-Blooming Club+6Late-Blooming ClubA Seed Begins to Stir+8A Seed Begins to StirThis Used to Be a Branch+10This Used to Be a BranchA Forest in a Bud+12A Forest in a BudThousand-Year-Old Tree+14Thousand-Year-Old TreeTree of the Wellspring+16Tree of the WellspringTree That Holds the World+18Tree That Holds the WorldYggdrasil, the World Tree+20Yggdrasil, the World Tree
Fishblades
Kelp-Wrapped Blade Shard+0Kelp-Wrapped Blade ShardCuttlebone Knife+2Cuttlebone KnifeCrab-Shell Sawblade+4Crab-Shell SawbladeFrozen Tuna+6Frozen TunaFish Skewer+8Fish SkewerGrilled Fish Skewer+10Grilled Fish SkewerPufferfish Mace+12Pufferfish MaceCoral Altar Blade+14Coral Altar BladeWhale Jawblade+16Whale JawbladeKraken Tentacle+18Kraken TentacleLeviathan's Fang+20Leviathan's Fang
Funerary Swords
Rusty Coffin Nail+0Rusty Coffin NailShroud-Wrapped Grave Knife+2Shroud-Wrapped Grave KnifeGravedigger’s Shovel+4Gravedigger’s ShovelBroken Tombstone Greatsword+6Broken Tombstone GreatswordCoffin-Lid Greatsword+8Coffin-Lid GreatswordBell That Wakes the Dead+10Bell That Wakes the DeadTorch of Wandering Souls+12Torch of Wandering SoulsGravekeeper’s Scythe+14Gravekeeper’s ScytheUprooted Grave+16Uprooted GraveGraveyard of the Dead+18Graveyard of the DeadKey to the Underworld+20Key to the Underworld