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Gen I (R/B)
Gen III (FR/LG)
Apart from evolution stones, there are several directly usable evolution items. All of these have been introduced in Generation VIII or later. As the Everstone can only be held in Pokémon Sword/Shield and Pokémon Scarlet/Violet, we will only focus on the ones available in those games.
These items are: 
Sweet Apple, 
Tart Apple, 
Syrupy Apple, 
Cracked Pot, 
Chipped Pot, 
Galarica Cuff, 
Galarica Wreath, 
Scroll of Darkness, 
Scroll of Waters, 
Auspicious Armor, 
Malicious Armor, 
Metal Alloy, 
Unremarkable Teacup, and 
Masterpiece Teacup.
Additionally, there are some new and obscure evolution methods, such as White-Striped Basculin, Primeape, Gimmighoul, Finizen, Tandemaus, Bisharp, and Pawmo/Rellor. Hisuian Qwilfish's evolution falls under "leveling up while knowing a certain move" (Barb Barrage) and does not activate if leveled up while also holding an Everstone:
These effects were tested on the following versions. Pokémon Violet was only used to test the Malicious Armor Charcadet evolution.
The following video demonstrates all Sword and Shield direct-use evolution items being unusable while the Everstone is held and will be referenced in the next sections:
Starting in Pokémon Sword/Shield, Rare Candy can be used to evolve Pokémon that have reached level 100 if their condition for level-up evolution is met. This evolution can be canceled by pressing B, and doing so will waste your Rare Candy.
If the Pokémon is holding an Everstone, the Rare Candy cannot be used and will instead say "It won't have any effect."
Rare Candy behavior with Everstone is consistent across Sword/Shield, Brilliant Diamond/Shining Pearl, and Scarlet/Violet.
In Pokémon Sword/Shield and Brilliant Diamond/Shining Pearl, attempting to use an evolution stone on a Pokémon holding an Everstone will display "Incompatible" and say "It won't have any effect" when used.
Unlike in Pokémon Sword/Shield and Brilliant Diamond/Shining Pearl, in Scarlet/Violet, evolution stones can be used when the Pokémon is holding an Everstone:
The Everstone is not consumed after evolution, but it is internally temporarily removed to facilitate evolution. For this reason, unlike any other (non-evolution facilitating) held items, it disappears from the Pokémon's sprite after returning to the Bag menu after evolution, but it will reappear if refreshed. (That is, the Pokémon is still holding the Everstone after it evolved). This is also the case for all other directly usable evolution items in Sc/Vi.
The video below demonstrates that the visual bug does NOT occur if the held item is anything other than the Everstone:
Applin evolves into one of its three evolutions (Flapple, Appletun, and Dipplin) depending on which of the three Apple items it is exposed to (Tart Apple, Sweet Apple, and Syrupy Apple respectively).
In Pokémon Sword/Shield, the Sweet and Tart Apples cannot be used to trigger an evolution when the Applin is holding an Everstone.
In Pokémon Scarlet/Violet, the apples can be used directly and cause evolution even when the Everstone is held:
Sinistea's Phony Form evolves when exposed to a Cracked Pot, while its Antique Form evolves when exposed to a Chipped Pot. Similarly, Poltchageist's Counterfeit Form evolves when exposed to an Unremarkable Teacup, while its Artisan Form evolves when exposed to a Masterpiece Teacup.
In Pokémon Sword/Shield, the pots cannot be used to trigger an evolution when the Sinistea is holding an Everstone.
In Pokémon Scarlet/Violet, all four Pokémon are able to evolve when their respective evolution items are used on them, even if they are holding an Everstone.
Galarian Slowpoke evolves into Slowbro when exposed to a Galarica Cuff, and into Slowking when exposed to a Galarica Wreath.
In Pokémon Sword/Shield, these items cannot be used to trigger an evolution when the Galarian Slowpoke is holding an Everstone.
In Pokémon Scarlet/Violet, the Galarica Cuff and Galarica Wreath can be used on Slowpoke to evolve it even if it is holding an Everstone: (both were tested, but only the video for Galarica Cuff is included)
In Scarlet/Violet, the scrolls used to evolve Kubfu became items (rather than interactable objects in the overworld). These items can be used on Kubfu from the bag to evolve it. Like the other items, using these items can evolve Kubfu even if Kubfu is holding an Everstone.
Charcadet evolves into Armarouge when exposed to Auspicious Armor or into Ceruledge when exposed to Malicious Armor. In Scarlet/Violet the armors are version-exclusive. Both armors can be used on Charcadet to evolve it even if it is holding an Everstone.
Duraludon evolves into Archaludon when exposed to Metal Alloy. Again, like the rest of the items, this evolution can occur even if Duraludon is holding an Everstone.
White-Striped Basculin evolves into Basculegion if it levels up after losing at least 294 HP from recoil damage without fainting. (Healing through items or at a Pokémon Center is allowed)
Primeape evolves into Annihilape when leveled up after using Rage Fist 20 times. Like other level-up evolutions, this evolution will not occur even if the conditions are met if the Everstone is held by Primeape when it levels up.
Gimmighoul evolves when leveled up with 999 Gimmighoul Coins in your Bag. This evolution also consumes those coins and cannot be canceled with B. If leveled up while holding an Everstone with 999 Gimmighoul Coins in your Bag, Gimmighoul will not evolve and the coins will not be consumed.
Finizen evolves into Palafin when leveled up while in a Union Circle starting from level 38.
I have not tested this yet, but I assume that like other level-up evolutions, it will not occur if Finizen is holding an Everstone.
Tandemaus evolves if it was leveled up starting from level 25 while in a battle. It is the only Pokémon with such a restriction. If it levels up in battle while holding an Everstone, it will not evolve. In the scenario of the video below, Tandemaus would have evolved with an evolution animation after the battle ended had it not held an Everstone.
Bisharp evolves into Kingambit if it is leveled up after defeating three other Bisharp holding a Leader's Crest (this held item is only held by Bisharp who spawn with a group of Pawniard). Like other level-up evolutions, this evolution will not occur even if the conditions are met if the Everstone is held by Bisharp when it levels up.
These Pokémon are included together because their evolution method is the same. Pawmo and Rellor both evolve into Pawmot and Rabsca respectively when leveled up while outside of its Poké Ball after the player walks 1000 steps with it using the Let's Go! feature. In the included video, it is demonstrated that leveling up while holding an Everstone does not trigger this evolution method even if the condition is met. (Leveling up immediately after removing the Everstone triggers the evolution)
Directly usable evolution items and evolution stones cannot be used on Pokémon holding an Everstone in Pokémon Sword and Shield, but CAN be used on a Pokémon holding an Everstone in Pokémon Scarlet and Violet. (doing so temporarily removes the Everstone internally and results in a minor visual glitch, see "The Everstone is not consumed ..." paragraph in the "Evolution stones in Scarlet/Violet" section above). Presumably, all evolution methods are disabled when the Everstone is held (which is why the game would need to remove the Everstone temporarily when directly usable evolution items are used, as it doesn't do that for any other held item), so a level 100 Pokémon holding an Everstone whose evolution condition is met also cannot be evolved by Rare Candy.
All evolution methods in Generations VIII and IX (other than directly usable items in Sc/Vi only and the trade evolution of Kadabra) cannot occur when the Everstone is held.