Well, lemme say a few formations I know about.
Sonic Stream = 11 successful Aya Scans on monsters lower level then her (Just go back to stage 1/2/3 and do it easy), then go to a healing circle (Used Hakurei Shrine's, don't think it's relevant.)
-It says that this formation always lets you move first. Didn't test it though.
Poison Shield = Something to do with killing stuff with Deadly Poison. You might need to beat Stage 8 boss before you can get it . Just have Byakuren use Hyper Trigger on Satori, and have her use Poison Art on an enemy encounter that is completely vulnerable, and defend until they die. Go to a healing circle after you do this to get it; I personally used the Sunflower Field one but I don't think it's relevant which.
-This formation is actually pretty cool. One spot in the front gets "amazing defense", while everyone else is in the rear. In other words, put Alice in the front and have her spam Parrar to shield the entire party. Also protects the party from poison.
Inferno/?
= Use Satori's Last Word 10 times (First formation) and 25 times(Second formation), then go talk to Okuu/Orin. I'm at about 15 uses so I don't know the second one.
-Inferno boosts everyone's attack, lowers their defense, and increases Fire damage and resistance. Iunno about the second one, as I haven't gotten it yet.
Shinigami Formation = I think this requires 10 total party wipes, but I'm not sure. Go back to Komachi after you defeat her and poke her, and you'll get it. If this is the 10 total party wipe formation, then I think poking Komachi again after 100 wipes might get you another one, but I'm not sure. That might be Tenshi you're supposed to poke for 100.
-Raises everyone's resistance to instant death. Nothing special.
some formation about the devil, whatever = Kick Meiling a whole bunch of times after you get Sakuya. Maybe it was 256. Just hold A for awhile, this isn't as bad as it sounds, but you should probably eat something while you hold the button anyway.
-Useless, this is a 2 person formation. Unless you have 2 people wiping one boss in the triple-split stage, you're never going to use only 2 people, now are you?
Those are the formations I remember and have gotten. Not sure how to get anything else, apart from the automatically obtained ones.