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The following content contains subjective advice or recommendations. Please take it into consideration and make your own decisions based on your playstyle and goals.
What are the best skill mons?
Not all skill Mons are created equal. Species aside, we must take a look to see if their main skills are viable.
List of Main Skills
This is the list of all the skills in the game. Not all of them are meta-relevant so I will group them based on role and focus on just the ones that are. Note that some skills are variants of each other.
The following are what are often considered the most meta-defining skills in the game.
Energy For Everyone (E4E/Healer)
This is THE most meta-defining skill in the game. It allows your team to be fully topped up on energy throughout the day. This is significant because maintaining energy above 81% allows your Mons to work 2.2x faster. The best healers currently are Gardevoir and Pawmot—Gardevoir being slightly better. With the changes to New Player AB, it is recommended to forgo Wigglytuff and Sylveon as temp healers since SDT and LL are now quicker to unlock.
Tasty Chance S
TC is a powerful late game skill that increases cooking crit chance. It is arguably the second most meta-defining skill in the game—providing a significant boost to Strength per meal. Dedenne is the premiere TC Skill Mon with honorable mentions to Cramorant, Weavile, and Swampert for being non-skill TC Mons. Players should consider Tasty Chance on these Mons as a bonus rather than the main focus. Always build around their respective Specialties first and foremost.
Cooking Power-Up (Pot Mon)
Another component necessary for late game cooking strats, these Mons enable access to late game dishes consisting of 100+ ingredients without use of a GCT. The max space a Potmon can add to your next meal is +200. The best Potmons are Magnezone followed by Toxtricity Low Key, then Flareon/Glaceon.
Ingredient Magnet (Ing Mag)
IM is better suited as an early-to-mid game option to kickstart cooking since cooking consistently can be difficult prior to leveling Ingredient Mons to 30. Despite that, it remains a useful supplement as you continue to build out your arsenal of Ingredient Mons (which may take a year or longer to complete). The best IM Mons are Vaporeon, Heracross, and Plusle. Though, Heracross is expensive and rare, and Plusle wants another plus/minus mon on the team so they are less recommended.
Charge Strength, Berry Burst (Strength Mons)
The purpose of these mons is to provide Snorlax Strength, meaning any way that benefits Strength gains is a boost to these mons. That means subskills like BFS, HB, and Helping Speeds (when BFS is involved) are better to have on these mons over Skill Triggers. They compete directly with BFS berry mons on their respective islands so they need to be judged much more harshly due to competition. Ironically, a trigger-based Charge Strength mon, despite having a lower strength ceiling than a BFS-based hybrid, has an early game niche of being able to provide lots of strength through their skill before berry mons can beat them due to BFS mons needing levels to get to higher strengths.
Dream Shard Magnet
DSM Mons are covered extensively in Pokémon Sleep Discord server's #the-swa-lot pins. I personally believe this is the 4th or 5th most important skill mon in the game as they really help you push multiple mons to lvl 60+.
Legendary Skills
Legendary Mons can provide solid value. However, these Mons are extremely niche due to their high Biscuit catch cost, meh default stats upon initial encounter, highly specific team building, and low encounter rate outside of dedicated events. In no way are Legendaries mandatory for game progress.