Here's a paradigmatic problem of the many (Geach and Unger are the usual sources cited, but I'm not claiming this to be exactly the version they use.) Let's take a moulting cat. There are many hairs that are neither clearly attached, nor clearly unattached to the main body of the cat. Let's enumerate them 1---1000. Then we might consider the material objects which are the masses of cat-arranged matter that include half of the thousand hairs, and exclude to the other half. There are many ways to choose the half that's included. So by this recipe we get many many distinct masses of cat-arranged matter, differing only over hairs. The various pieces of cat-arranged matter change their properties over time in very much the way that cats do: they are now in a sitting-shape, now in a standing-shape, now in a lapping-milk shape, now in an emitting-meows configuration. They each seem to have everything intrinsically required for being a cat.
If you're inclined to think (and I am) that a cat is a material object identical to some piece of cat-arranged matter, then the problem of the many arises: which of the various distinct pieces of cat-arranged matters is the cat? Various answers have been suggested. Some of the most obvious (though not necessarily the most sensible) are: (i) nihilism: none of the cat-candidates are cats. (ii) brutalism: exactly one is a cat, and there is a brute fact of the matter which it is; (iii) vague cat: exactly one is a cat, and it's a vague matter which it is; (iii) manyism: lots of the cat-candidates are cats.
(By the way, (ii) and (iii) may not be incompatible, if you're an epistemicist about vagueness. And those who are fans of many-valued logics for vagueness should have a think about whether they can really support (iii). Consider the best candidates to be a cat, c1....c1000. Suppose these are each cats to an equal degree. Then "one of c1...c1000 is a cat" will standardly have a degree of truth equal to the disjunction=the maximum of the disjuncts=the degree of truth of "c1 is a cat". And the degree of truth of the conjunction: "all of c1...c1000 is a cat" will standardly have a degree of truth equal to the conjunction=the minimum of the conjuncts=the degree of truth of "c1 is a cat". So to the extent that the (determinately distinct) best candidates aren't all cats, to exactly that extent there's no cat among them (and since we chose the best candidates, we won't get a higher degree of truth for "the cat is present" by including extra disjuncts. Conclusion: if you're tempted by response (iii) to the problem of the many, you've got strong reason not to go for many-valued logic. [Edit (see comments): this needs qualification. I think you've reason not to go for many-valued logics that endorse the (fairly standard, but not undeniable) max/min treatment of disjunction/conjunction; and in which the many values are linearly arranged].)
What I'd really like to emphasize is the above leaves open the following question: Is there a super-cat-candidate, i.e. a piece of cat-arranged matter of which every other cat-candidate is a proper part? Take the Tibbles case above, and suppose that the candidates only differ over hairs. Then a potential super-cat-candidate would be the piece of matter that's maximally generous: that includes all the 1000 not-clearly-unattached hairs. If this particular fusion isn't genuinely a cat-candidate, then it's open that if you arrange the cat-candidates by which is a part of which, you'll end up with multiple maximal cat-candidates none of which is a part of the other. Perhaps they each contain 999 hairs, but differ amongst themselves which hair they don't include.
If there is a super-cat-candidate, let's say the problem of the many is of type-1, and if there's no super-cat-candidate, let's say that the problem of the many is of type-2.
My guess is that our description of cases like Tibbles leaves is simply underspecified as to whether it's of type-1 or type-2. But I certainly don't see any principled reason to think that the actual cases of the POM we find around us are always of type-1. There's certainly no a priori guarantee that the sort of criterion that rules in some things as parts of a cat won't also dismiss other things as non-parts. So for example, perhaps we can rank candidates for degrees of integration: some unintegrated parts are ok, but there's some cut-off where an object is just too unintegrated to count as a candidate. One cat-candidate includes some borderline-attached skin cells, and is to that extent unintegrated. Another cat-candidate includes some borderline-attached teeth, and is to that extent unintegrated. But plausibly the fusion that includes both skin cells and teeth is less integrated: enough to disqualify it from being a cat-candidate. It's hard to know how to argue the case further without going deeply into feline biology, but I hope you get the sense of why type-2 POM need to be dealt with.
Now, one response to the standard POM is to appeal to the "maximality" allegedly built into various predicates (like "rock", "cat", "conscious" etc): things that are duplicates of rocks, but which are surrounded by extra rocky stuff, become merely parts of rocks (and so forth). There are presumably intrinisic duplicates of rocks embedded as tiny parts at the centre of large boulders: but there's no intuitive pressure to count them as rocks. Likewise a cat might survive after it's limbs are destroyed by a vengeful deity, but it's unintuitive to think of the duplicate head-and-torso part of Tibbles as itself a cat-candidate. So there's some reasons independently of paradigmatic problem of the many scenarios to think of "cat" and "rock" etc as maximal. (For more discussion of maximality, see Ted Sider's various papers on the topic).
If we've got a type-1 problem of the many, then one might think that the maximality of "cat" or "rock" or whatever gives a principled answer to our original question: the super-cat-candidate (/super-rock-candidate) is the one uniquely qualified to be the cat (/rock). For we've then got an explanation for why all the others, though intrinsically qualified just like cats, aren't cats: being a cat is a maximal property, and all the rival cat-candidates are parts of the one true cat in the vicinity.
But the type-2 problem of the many really isn't addressed by maximality as such. There's no unique super-cat-candidate in this setup, rather a range of co-maximal ones. So maximality won't save our bacon here.
The difference between the two cases is important when we consider other things. For example, in the light of the (fairly widely accepted) maximality of "house" and "cat" and "rock" and the like, few would say that any duplicate of a house must be a house (even setting aside extrinsicality due to social setting). But there's an obvious fall back position, which is floating around the literature: that any duplicate of a house must be a (proper or improper) part of a house (holding fixed social setting etc). That is, any house possesses the property of being part of a house intrinsically (so long as we hold fixed social setting etc). And the same goes for cat: at least holding fixed biological origin, it's plausible that any cat is intrinsically at least part of a cat, and any rock is intrinsically at least part of a rock.
These claims aren't threatened by maximality. But appealing to them in a type-2 problem of the many gets us an argument directly for response (iv): manyism. For plausibly if you took a duplicate of one of the co-maximal cat candidates, T, while eliminating from the scene those bits of matter that are not part of T but are part of one of the other co-maximal cat candidates, then you get something T* that's (determinately) a cat. And so, any duplicate of T* must be at least part of a cat. And since T is a duplicate of T*, T must be at least part of a cat. But T isn't proper part of anything that's even a cat-candidate. So T must itself be a cat.
So the type-2 POM is harder to resolve than the type-1 kind. Maybe some extra weakening of the properties a cat-candidate has intrinsicality are called for. Or maybe (very surprisingly) type-2 POMs never arise. But either way, more work is needed.