This section is for cards that can move cards from the discard pile to the hand or deck.
(This is just me, but I refer to discard → hand as "recovery", and discard → deck as "recycling", after the Fossil/Emerging Powers card.)
For effects that attach Energy from the discard to Pokémon in play, see Energy Acceleration — Discard to Play.
The default recovery card. It's useful for getting back mid/late-game cards which you had to discard to Ultra Ball or similar earlier, or an attacker/Fezandipiti ex after it gets KOed.
2 Night Stretcher is a common sight in many decks. If you play 2 copies of a certain Pokémon and end up discarding 1, Night Stretcher makes it so you now have 3 outs to finding that Pokémon (2 Night Stretcher + the other copy in your deck).
Slower than Night Stretcher, but still sees plenty of play. Super Rod is useful for recycling multiple cards if you end up discarding them to Professor's Research, Ultra Ball, Squawkabilly ex, etc.
Putting Energy back into your deck is also useful if you have effects that accelerate Energy from your deck.
These cards are useful if you have attackers that discard Energy, e.g. Gholdengo ex or Raging Bolt ex. Crispin/Earthen Vessel/Energy Search Pro can power up the first few attacks by getting Energy from the deck, but the Energy will be in the discard pile after that, so it'll be (Superior) Energy Retrieval's job to fuel subsequent attacks.
XY Block taught us that VS Seeker cannot exist in the same format as Professor's Research, so this is what we have instead. Pal Pad functions similarly to VS Seeker, letting you recycle 1-of Supporters, but you have to draw into or search for them again.
Despite this ostensibly being a support card for Iono's Pokémon decks, it ended up seeing play in a completely different archetype: Gholdengo ex. It supplements Superior Energy Retrieval in getting Energy from the discard pile back to hand for Make it Rain.
Triple Night Stretcher, but only for non-Rule Box Pokémon, and as a Supporter.
Only really playable in 1-Prizer decks. Or in Gholdengo ex, as a clunky 5th Superior Energy Retrieval that might be able to recover Scizor against walls.
Quintuple Night Stretcher, but is an ACE SPEC.
Played in Slowking to rebuild an attacking Slowking after it gets KOed. It'll also let you reuse expended attackers by returning them to hand and putting them on top of the deck with Academy at Night.
A rarely seen card. It's really only playable in 1-Prizer decks (e.g. Ethan's Typhlosion), where it can recycle multiple Evolution lines, and the 1-Prizer nature means you bleed Prizes slowly enough that you can actually draw/search into the cards that you recycled again.
Super Rod is generally better. You'd have to shuffle 4-5 Pokémon back in for Sacred Ash to outperform Super Rod.
Sacred Ash's counterpart that recycles Basic Energy instead of Pokémon.
It's even rarer than Sacred Ash, and only playable in decks that accelerate Energy from the deck, like Yanmega ex.
Only seen in control decks, to reuse resources that would otherwise be impossible to recover.
Slime Mold Colony blocks quite a few recovery cards. However, it will not block effects that shuffle cards from the discard pile into the deck (Super Rod and Pal Pad), and it will not block Professor Sada's Vitality (because the cards get attached to Pokémon, not put into the player's hand).