Nest Ball

The default Basic searcher. There are no restrictions on what it can get, just that the search target has to be a Basic.

Putting Fezandipiti ex in your deck means Nest Ball is live late-game too, so you can play 4 copies of Nest Ball with confidence.

Buddy-Buddy Poffin

A staple Basic searcher for decks that use Evolutions. The Basic forms are often 70 HP or less, and Buddy-Buddy Poffin gets two of them.

Artazon

A supplementary Basic searcher for decks that use Evolutions. As a Stadium, it can be used even if the opponent has you under Item lock, but note that your opponent can use it too.

Artazon is a common search target for Secret Box, converting the Stadium search into a Basic search.

Fan Rotom

Fan Call searches for the Basic forms of several draw/search engines: Hoothoot, Pidgey, Dunsparce. Notably, it can also get Noctowl as the follow-up to Hoothoot, as well as Ditto.

Ditto

A pseudo-Nest Ball, but slightly complicated by the "first turn only" and "must be Active" requirements for its Ability.

Common partners for Ditto are Fan Rotom searching it out, and Latias ex to retreat your starter into it for free.

Miraidon ex

A repeatable double Nest Ball, but for Lightning-types.

It can get both acceleration (Joltik, Magnemite, Tynamo) and an attacker (Iron Hands ex, Pikachu ex, Zekrom ex) into play off one search.

Precious Trolley

All-you-can-eat Nest Balls, limited only by your Bench space. Card advantage wise, Precious Trolley can go as high as +4, or even +7 if Area Zero Underdepths is in play and you have only one Tera Active.

Pidgey

This pre-evolved form of Pidgeot ex has a utility attack that Benches 2 Basics from your deck.

Bombirdier ex

This is primarily meant as a way to beat Cornerstone Mask Ogerpon ex, but it can set up if you happen to start with it. It mimics Dunsparce SS/CES, with an attack that Benches 3 Basics from your deck. It even imitates the EX era rule of being able to attack if you go first.

Ultra Ball

The default Evolution searcher, almost every deck uses it. It's a -2 in card advantage, but draw Supporters and Abilities can refill your hand.

Hilda

Getting a single Evolution alone is not impressive, but Hilda combines that with the rare ability to search for any Energy (not just Basic Energy), meaning it can get Luminous Energy for Munkidori if your deck normally wouldn't be able to splash Basic Darkness, as well as powerful ACE SPECs like Neo Upper Energy.

Cyrano

In decks with Stage 1 ex's (e.g. Flareon ex, N's Zoroark ex), Cyrano is a better Jacq, so much so that you can play it in multiples.

Note that Cyrano can search for Basics too, as long as they are ex (e.g. Fezandipiti ex, Eevee ex).

Genesect ex

Metallic Signal searches for 2 Metal-type Evolutions.

Played in Gholdengo ex. It can search for Scizor.

Tera Orb

Searches for a Tera Pokémon.

Generally only played in decks with Tera Evolutions, prominently Flareon ex. While you can search for Tera Basics with Tera Orb, you already have Nest Ball for those.

Mesagoza

Poké Ball as a Stadium.

Played in the Secret Box build of Charizard ex as a risky way to convert the Stadium search into a valuable Evolution Pokémon.

Hyper Aroma

Very rarely played. You might want it if you have a thick 4-4(-X) Evolution line where the Stage 1 has draw/search Abilities, so after getting the Basics down, Hyper Aroma can pull out 3 Stage 1s and give you 3 activations immediately.

Earthen Vessel

The default Energy searcher. Helps fix colors if your deck runs 2 or more types of Basic Energy.

The Ancient type on it is trinket text. You don't need any Ancient synergies in your deck to play Earthen Vessel.

Energy Search Pro

Played exclusively in the most Energy-hungry deck, Gholdengo ex. Energy Search Pro can get as many as 8 Basic Energy (if none of your 1-ofs are Prized).

Energy Search

An infrequently played card that can convert an Item search from Arven or Secret Box into an Energy search.

Charizard ex or Marnie's Grimmsnarl ex plays it sometimes, as these decks want to retain Energy in the deck for their Abilities, so getting 1 Energy off Energy Search is preferable to 2 off Earthen Vessel. In Charizard ex, it can also get the 1-of Darkness Energy for Munkidori.

Letter of Encouragement

Only mentioned here for historical reasons. It was played in Gholdengo ex before Energy Search Pro.

Noctowl

Noctowl searches for 2 Trainers when played from hand. Note that you need a Tera Pokémon to use this effect.

Best friends with Fan Rotom. These one-use support Pokémon can take up a lot of Bench space, but Area Zero Underdepths (which is searchable off Noctowl) increases the amount of room you have.

Secret Box

The "5th Arven". Secret Box can get the Buddy-Buddy Poffin + TM: Evo combo, just like Arven, plus Artazon for one more Basic, and Iono or Professor's Research to refill your hand after vomiting it out onto the board.

Of course, other combinations are possible; it depends on what's in your deck and what you need at that moment.

Colress's Tenacity

With the right Stadiums in your deck, Colress's Tenacity can be converted into a Basic (Artazon), 50-50 Pokémon (Mesagoza), or Pokémon Tool (Town Store) search, or a damage buff (Practice Studio, Perilous Jungle, Gravity Mountain).

Like Hilda, Colress's Tenacity can get Special Energy cards.

Note that Colress's Tenacity will let your opponent's Kyurem use Trifrost for {C}, though that card is very rare and often just fodder for another Pokémon to copy anyway.

Town Store

This converts the Stadium search from Colress's Tenacity into a Pokémon Tool search.

Team Rocket's Petrel

Fringe playable without the Team Rocket Engine. It is inferior to Arven for the most part, searching 1 fewer card, but it can get Stadiums like Spikemuth Gym (to act as a psuedo-Brock's Scouting or Jacq) or Team Rocket's Watchtower (for disruption).

Pidgeot ex

Pidgeot RG returns, but as a 2-Prizer in a format with nerfed Rare Candy (both in terms of the card's own functionality, and the presence of Budew suppressing it), and Boss's Orders to drag it out for KOs. It's still a decent support Pokémon, primarily with Charizard ex, but is no longer a staple for Stage 2 decks.

Ciphermaniac's Codebreaking

Retro players will recognize this as identical to Oracle. Just like how Oracle was paired with Delcatty in the 2004 format, you'll need to pair Ciphermaniac's Codebreaking with a draw Ability (e.g. Gholdengo ex, N's Zoroark ex) to get the cards immediately.

It can also set up an attacker on top of the deck for Slowking's Seek Inspiration.

Thwackey

Festival Lead's Pidgeot ex. It has many advantages over Pidgeot — 1-Prizer, Stage 1, not hard once per turn — but it only goes into one specific archetype.