To evolve a weapon you need to fulfill those conditions prior to level up:

• Weapon has to be lv 10
  
• You need to have one crown (drop from red glowing elite mob). Important note: crowns have a chance to spawn ONLY when at least one weapon is level 10 and not yet evolved

You can see how many crowns you have on the HUD with the crown icon below your life bar. It is the best indication as crowns are really hard to see among all the drops and are collected automatically at the end of a wave.

Next time you level up you will automatically have the choice to evolve all eligible weapons, for instance if you have two level 10 weapons and two crowns, next level up will present you the choice to evolve those.

Important: the evolve choice is guaranted to pop up with each level up when possible, so you are not obligated to evolve the first time it shows, you can wait in order to priorize another weapon/or the special evolution if you want.

Some weapons have two possible evolutions:

• the normal one described before
  
• the special one requires to have one particular talent maxed (lv 5) prior the time you are offered evolution. Special evolution has a different name than the original and is listed in the Compendium.

If you fullfill the additional talent requirement for eligible weapons
at next level up you will be offered two guaranted choice: normal evolution and special evolution

Here is an exemple of level up screen: in that case the snickerdoodle is lv10 and I have vamp spreading talent at lv 5 , I also have 3 crowns and evoluion require one, so I have the choice between normal (at right) and special evolution (left, Resonant Ringburst), there is no advantage in picking normal since special is more powerful and has the same crown cost.

Once you've done them once their entries will unlock in the Compendium

NEW EVOLUTIONS version 1.1.17
A Blacker Death
Backstab Bounty
Claw Shield
Cursed Cranium
Traverse
Mutagenic Menace
Silver Bind
Sinister Symbiosis
Solar Storm
Spectral Bat Blitz
Stake Cascade
Stone Cold
Summoned Snare
Symphony’s Surprise
Trident
Vital Hits
Winged Vortex

Notes:
Devouring Tome is finally obtainable as Slayer Talent is properly unlockable now (Devouring Tome = Self help book + slayer)

Among the non characters specific weapons only duelling pistol, marble cat and refectory table have no special evolutions

Among the characters Mark, Renfield, Rebecca and Robert have a special evolution for their specific weapons,

Thanks to Mega Cat Studios for this awesome game.
Thanks to sophialeigh for their help finding the evolutions.