I played around with this quite a bit; here are some of the things I found. I wasn't testing if it was possible to get out of any of this; these are just the things that looked best against a standing AI in training mode. I used Bowser for most tests since he's huge and easy to footstool, but as King Dedede demonstrates, that means I very likely missed some character exclusive stuff.
Mario and Luigi gets absolutely nothing.
Peach can float out of footstools so she obviously gets down aerial. She can hit with the Parasol too, but I'm not sure why you would want to. This might be a good way to set up to hit with the super turnip or a bomb.
Bowser doesn't get anything really, but if they have a low shield, Bowser Bomb could be a good choice (that move eats shields for those who forgot).
All DK gets is his forward special, but why would you want anything else anyway?
Diddy Kong can't land any aerials, but he can fire off his Rocketbarrels for a solid 10% hit. I suppose you could also chuck a banana downward.
I think Yoshi gets Ground Pound for sure out of this (and it looks like he can hit with Egg Roll... though it's useless), but he can't hit with aerials.
Wario gets nothing... not even a full Waft works.
Link can throw a bomb down but otherwise nothing.
Zelda only gets Nayru's Love. It looks like Sheik gets down aerial out of it but nothing else.
Ganondorf seems to get Wizard's Foot which is some pretty nice damage. I don't think he gets anything else.
Toon Link gets that lovely down aerial and can drop a bomb of course.
Samus doesn't get anything, but footstool to bomb does kill her momentum and seems like it could be handy for pressure.
Zero Suit Samus gets down aerial and seemingly nothing else excepting the obvious armor piece toss.
Pit doesn't land anything normally, but the Wings of Icarus totally kill his momentum and seem like they set him up for pretty much any aerial he wants. Pit players would probably know more about how to take advantage of that. Also, for the record, you can't glide out of a footstool.
As expected, the Ice Climbers just get the down aerial. This could be pretty cool with desynching I bet.
R.O.B. gets his down aerial and seemingly nothing else.
Kirby's hammer is probably best if you can do the pulling back stunt DK can, but stone seems to just barely not hit fast enough to combo.
Meta Knight only gets down aerial as has been claimed, but Dimension Cape is a funny choice (I'm pretty sure it doesn't combo).
King Dedede seems to do better with down aerial against some characters than others. He doesn't hit Bowser at all, hits Meta Knight and Snake just once, and hits Captain Falcon a bunch. None of his other stuff seems like it works.
I'm pretty sure Olimar gets nothing.
Fox's shine and illusion both cancel his momentum, but both hit too high. Fire Fox looks like it would give a very small window to escape since it sits above not hitting for a moment, but it is probably the best option.
Fire Falco, on the other hand, is not even close to worth it. On the other hand, Falco Phantasm actually does hit.
I don't think Wolf gets anything worthwhile.
Falcon Kick looks very forced, but nothing else Captain Falcon has seems to work.
Pikachu can Quick Attack right out of a footstool which I'm sure those Pikachu mains can put to devious use. Headbutt also cancels his ascent, but I'm not sure if it's even possible to hit with it (not that it's a good move anyway).
Squirtle seems to get a very nice hit out of Waterfall. Ivysaur seems to just get the down aerial (not that that's bad at all). Charizard seems to be able to combo into Fly which is VERY damaging. This is great for the Pokemon Trainer!
Predictably, Lucario seems to get his down aerial. Double Team and Extremespeed kill his momentum, but that's pretty useless for obvious reasons (I suppose you could have an epic moment where you footstool Popo and cancel with Double Team to nail Nana or something...).
Jigglypuff gets off a down aerial out of this. If I recall, down aerial can trip and lead into Rest with her so that's kinda cool.
Marth can cancel his ascent with Counter and otherwise has nothing of note.
Ike continues the trend of useless things that cancel the ascent with Aether (which doesn't hit) and Counter.
Ness can cancel his ascent with Pk Thunder and Pk Flash, but neither seem like they acutally are conducive to anyone but Ness getting hit.
Lucas's big story is still the down aerial, but he has the useless options of Pk Thunder and Pk Freeze just like Ness.
Mr. Game & Watch only gets the key here. I was really hoping it would combo into Oil Panic since the bucket cancels the ascent, but it only does when it's not full.
Snake could drop a grenade he was already carrying on them or hit with just the first hit of his down aerial.
Sonic gets his down aerial or his spring here.