1: Fox, Falco, Marth, Sheik, Puff, Peach
2: Falcon, Pika, ICs, Samus, Yoshi, Luigi
3: Doc, Mario, Ganon, Young Link, Mewtwo, DK
4: The rest.
The focus is mostly on the tiers, not the ordering within tiers.
Comments on tiers:
1: We know they can win a major. Demonstrated top tier tournament results.
2: Some of these characters can win a major. We aren't totally sure which.
3: Harder to win with. Can win a smaller tournament.
4: Bad enough I'd recommend you avoid them. Pichu is underrated but still bad. Zelda and Roy are overrated. Link and Game & Watch aren't good enough. Yes, you could still win a tournament with any character if you're significantly better than the other players.
I'm not good enough or experienced enough at the game to know what I'm talking about that way, but I'm good at taking top players' explanations and judging which make sense.
I'd generally recommend people play tier 1 or 2 characters.
Peach and Puff are top tier because of proven tournament results from Armada and Hbox. Falcon looks like he should be a competitive character, and has put up some promising results, but he doesn't have the track record of the top tiers. The rest of tier 2 ranges from similar to Falcon to being more speculative.
I drew the line at Luigi, ahead of his clones, because having the best wavedash is more promising than a simliar character with worse mobility.
I'm not a big ICs fan because I place a high value on neutral game (getting the first hit in an even situation) and mobility. ICs has very strong punishes (e.g. wobbling) and is a skill test for opponents (if they mess up, they lose). That doesn't work well at the highest level and it isn't fun because it makes the other player control the outcome of the game more than you. I also don't like the idea of babysitting the computer-controlled character (that sometimes plays very stupidly) and playing a low tier when the other climber is dead. But fighting a two on one, even with some limits, is really powerful. So ICs do have potential.
And punishes do matter. If you win neutral 30% of the time for 40 damage on average, and lose neutral 70% of the for an average of 15 damage, then you're coming out ahead (120 damage to 105 per 10 engagements). And everyone does make mistakes. You can't play perfectly or react to everything.
Falcon has strong punishes but also has a strong neutral game due to amazing mobility.
Peach has some strong punishes but also has float cancels and turnips for neutral. Her problem is being a little slow.
Yoshi I don't understand very well. Partly, to me, he looks overly focused on his punish game. But he has parrying and super armor on his double jump that are unique and powerful tools, and I don't really know how good they are.
Luigi has a decent, viable moveset along with the best wavedash. The mobility gives him potential.
Samus has range, defense, wavedash back, projectiles, and recovery. But she's slow.
Puff has bair, horizontal air mobility, rest, and 5 double jumps. Puff would be bad without the large hitbox on bair that goes way beyond her foot. That one move really increases her range and zoning ability.
Fox and Marth have the best neutral games at the top level. Falco has the best neutral game at lower level where people struggle with his lasers. Marth sometimes struggles to get kills and has to win neutral a lot, and he has bad defense. Fox and Falco have the best offense but can be super vulnerable if you hit them.
Sheik has tools for neutral game (some speed, needles, auto cancel aeriels), edge guards, strong defense, strong grab punishes, and generally strong low-lag moves.
Dear new players: Fox's edge is small, don't worry about it, the other top tiers are fine too. At the start, pick any character from tier 1 or 2 that you like (or tier 3 if you really like them) and play that character until you're a halfway decent player who understands what's going on. Then reconsider what character you want to play once you actually understand what all the characters do and have a feel for playing the game.
Impatient players should choose Fox or Falco. Shine gives them the most aggressive options to do a variety of safe approaches and shield pressure. Or, better yet, learn to play patiently. Playing aggressively can be good if done in the right way, but charging in all the time isn't optimal on any character.
Players who aren't really hardcore gamers should not choose Fox or Falco. This is a very hard game with every character. Fox and Falco are a level above in terms of difficulty, and you really don't have to play them.