Luigi > Ice Climbers. I've played this match-up a bunch. On the surface it seems like Luigi's attacks are fairly easy to defend. His primary offenses are SH double aerials, dsmash, and ftilt. All pretty easy to anticipate, right? Well, it's not as easy as it seems. Luigi can approach extremely fast, meaning that he keeps the threat of attacking alive at all times. If Ice Climbers don't shield his attacks, they take massive damage. Dsmash in itself separates Nana and Popo and is a combo starter, so Luigi gets a free chop, dair, or whatever he wants after it connects. He can even go after Nana, as Popo is in the air and helpless to assist her. Luigi's dsmash has even more perks than this, in fact. It hits on frame 5, meaning that if you try to shield it on reaction, Popo might successfully do so, but Nana's shield often won't go up in time, meaning she gets hit, so Luigi takes less punishment from Popo, since Nana isn't there to buff the counterattack. Moreover, Ice Climbers have pretty bad shields. You'd be surprised how quickly Luigi's double aerials munch away at their defenses. Luigi's dsmash takes advantage of this well, since it hits low. Even if Popo gets his shield up in time, it's not uncommon for him to get shield stabbed by the dsmash.
Luigi's double aerials are a great tool at his disposal, for pressuring and weakening ICs' shields, for forcing openings and mistakes, for simply racking up damage. SH dair -> nair is particularly effective, as it takes advantage of the fact that dair is the only aerial that ICs can't duck when it's introduced at SH height. It can typically be followed up with a late nair on their shields, which has enough shieldstun to make shieldgrabbing extremely difficult. Luigi shouldn't have trouble spotdodging the attempted shieldgrab or dsmashing in similar wise as spacies shine to counteract it. This is devastating, as ICs cannot simply shield the dsmash, because their shield is a dot at this point. They'll undoubtedly get shield stabbed if they opt to do so. They cannot shieldgrab for the reason I mentioned. Nor can they roll, as Nana will get hit during the vulnerable frames of her roll. Therefore, once Luigi initiates his sequence of double aerials, ICs are more or less sunk. The only situation where Luigi's double aerials can really be challenged is when he hits the back of their shields. In that case, bair oos can fit in-between Luigi's aerials.
To make the battle even more inauspicious for ICs, Luigi cannot be dthrow dair chaingrabbed. It doesn't even "kind of" work, like on Marth or Peach. This means that the risks involved in grabbing Luigi are more pronounced, whereas in the case of a spacy or Sheik, getting a grab is worth the world. ICs can, of course, KO Luigi as any other character: grab, random smashes, and bair, although, in order to land these moves, there's the question of how to get within proximity of Luigi. Sure, you can wait for him to approach you. But he will not always oblige, and eventually you must needs bring the fight to him. As it turns out, Luigi's ftilt is, like, designed to keep ICs out. To anyone who has not played a wavedash-based character and experienced wavedashing into Samus or Luigi's ftilt repeatedly: I don't recommend trying it. Anyway, Luigi has that method of stifling ICs' advances. Luigi is also great at KO ICs. His upsmash, fsmash, dair, fair, and nair are particularly good at this. He also can occasionally get KOs with up-b, dsmash, and bair. His aerials, specifically, are well suited for intercepting Popo's solo squall.
That's pretty much all I have to bring to the party on this match-up. If anyone disagrees on any points, I'd be glad to go into deeper discussion.
