Sheik unlocks Greninja confirmed!
Edit: Actually, Link additionally unlocks Simon Belmont if you complete Classic Mode with King K. Rool already unlocked. So here's the stats that we know of so far:
1. Mario: Unlocks Sonic first, and then Pokémon Trainer.
2. Donkey Kong: Unlocks Pokémon Trainer first, and then Bowser Jr.
3. Link: Unlocks King K. Rool first, and then Simon Belmont.
4. Samus: Unlocks Inklings first, and then Isabelle; unlocks Dark Samus on 2P.
4e. Dark Samus: Unlocks Cloud.
5. Yoshi: Unlocks Lucario first and then Chrom (?).
6. Kirby: Unlocks Ness.
7. Fox: Unlocks Captain Falcon.
8. Pikachu: Unlocks Villager first (presumably), and then Shulk.
9. Luigi: Unlocks R.O.B.
10. Ness: Unlocks Jigglypuff first, and then Pac-Man.
11. Captain Falcon: Unlocks Falco first, and then Zero Suit Samus.
12. Jigglypuff: Presumably unlocks Pac-Man.
13. Peach: Unlocks Falco first, and then Wolf.
14. Bowser: Unlocks Pokémon Trainer.
15. Ice Climbers: Unlocks King K. Rool first, and then Simon Belmont.
16. Sheik: Unlocks Greninja.
17. Zelda: Unlocks Robin.
19. Pichu: Unlocks Rosalina and Duck Hunt (not sure in which order yet).
20. Falco: Unlocks Peach first, then Daisy, and then Bowser Jr.
21. Marth: Unlocks Ryu first, and then Lucina, and then Roy and Ike.
21e. Lucina: Unlocks Ridley.
22. Young Link: Unlocks Palutena and Richter Belmont (again, not sure in which order yet).
25. Roy: Unlocks Dr. Mario.
27. Meta Knight: Unlocks Snake
28e. Dark Pit: Unlocks Diddy Kong
30. Wario: Unlocks Dark Pit and Mega Man (not sure in which order yet).
31. Snake: Unlocks Richter Belmont.
32. Ike: Unlocks Luigi.
33-35. Pokémon Trainer: Unlocks King Dedede first and then Rosalina.
37. Lucas: Unlocks Palutena.
38. Sonic: Unlocks Bayonetta.
39. King Dedede: Unlocks Sheik.
42. R.O.B.: Unlocks Mr. Game & Watch.
47. Wii Fit Trainer: Unlocks Incineroar.
48. Rosalina: Unlocks Greninja.
49. Little Mac: Unlocks Roy, Ike, and Olimar (not sure of the order here, either).
57. Shulk: Unlocks Isabelle.
59. Duck Hunt: Unlocks Ken.
60. Ryu: Unlocks Lucina, Ganondorf, Palutena, and Ken (again, not sure of the order).
60e. Ken: Unlocks Lucas.
61. Cloud: Unlocks Wario.
62. Corrin: Unlocks Lucas.
63. Bayonetta: Unlocks Little Mac.
64. Inkling: Unlocks Wii Fit Trainer.
65. Ridley: Unlocks Chrom.
66. Simon: Unlocks Young Link.
67. King K. Rool: Unlocks Ice Climbers first, and then Dr. Mario.
68. Isabelle: Unlocks Pichu.
Also worth noting as well that once all the primary and secondary conditions have been fulfilled, trying to unlock anybody else with that character will simply unlock the next character in line in their very primary path. And once everybody on said path is unlocked, they won't unlock anything else anymore unless they have a 2P unlock condition (meaning until characters like Inkling, Isabelle, Wii Fit Trainer, and Incineroar are all unlocked, Samus will not even bother to try to unlock Dark Samus on 1P).
Lastly, there seems to be some conditioning behind some possible unlockable fighters. Because somebody actively trying to unlock Chrom have found themselves unlocking other fighters beforehand, such as Roy, Ike, and Lucina. So this might suggest that Chrom simply won't even appear as an unlockable fighter until certain conditions are successfully met. Same thing might apply to other echo fighters, as well as fighters with a personal history with other fighters. So it wouldn't surprise me if the game forces you down certain paths before you unlock other fighters.
And that's all I have for the time being. It seems like most, if not all of the original 8 have primary, secondary, and 2P conditions for unlocking characters. And I think it's safe to say that a large portion of the unlockable fighters may have these additions as well. This could perhaps explain why many fighters like Mewtwo, Meta Knight, or strangest of all, Zelda and Pit have yet to be uncovered via Classic Mode. It's also worth noting too that people have yet to even play Classic Mode with the intent of unlocking characters with characters like Villager or Lucario, so a lot of my information remains incomplete.
I might limit my playtime of Smash Ultimate to academic research. But it seems that many fighters will have up to three possible fighters to unlock. And many other low priority fighters may in fact require some secondary conditions to be fulfilled before they can successfully be unlocked. This in turn could explain why characters like Bowser and Mewtwo have yet to appear, especially since both characters are the final boss in quite a few Classic Modes. Maybe you have to clear Classic Mode with every fighter that encounters these enemies as the final boss before they themselves become available. These are just theories, but considering just how low priority a lot of these fightesr are, it seems one will have to do a lot more than just clearing Classic Mode in order to unlock certain fighters.