When would a Buddhist want to attach?

When you say attach, the official term for it is upādāna (clinging, grasping) in both Sanskrit and Pali.

In MN 11, there are four types of clinging described:

"Bhikkhus, there are these four kinds of clinging. What four? Clinging to sensual pleasures, clinging to views, clinging to rules and observances, and clinging to a doctrine of self.

Taking the example of clinging to sensual pleasures (kama-upādāna):

Eating food for physical sustenance, rather than for enjoyment is an example in SN 12.63 that shows us that we should not abandon food, rather we should abandon clinging to the pleasure of food. Practising extreme austerities that involve self-starvation is discouraged in Buddhism.

Of course, eventually, all types of clinging and craving have to be overcome in order to become permanently free of suffering.

So, does this mean that we should never cling to anything?

Buddhism does allow skillful use of limited clinging, in order to overcome our bigger shortcomings. This usually applies to clinging to views (diṭṭhi-upādāna).

The first example is Dhamma (teachings of the Buddha) and the Right View. One may cling to the Right View, in order to progress on the right path towards Nibbana, but eventually, he needs to abandon clinging to all views. In MN 22, the Dhamma is described as a raft that takes one across the river of samsara to the other shore (Nibbana), but eventually, one must let go of the raft, in order to get onto the other shore (Nibbana).

The second example are the following views from AN 5.57:

“And for the sake of what benefit should a woman or a man, a householder or one gone forth, often reflect thus: ‘I am the owner of my kamma, the heir of my kamma; I have kamma as my origin, kamma as my relative, kamma as my resort; I will be the heir of whatever kamma, good or bad, that I do’? People engage in misconduct by body, speech, and mind. But when one often reflects upon this theme, such misconduct is either completely abandoned or diminished. It is for the sake of this benefit that a woman or a man, a householder or one gone forth, should often reflect thus: ‘I am the owner of my kamma, the heir of my kamma; I have kamma as my origin, kamma as my relative, kamma as my resort; I will be the heir of whatever kamma, good or bad, that I do.’

“This noble disciple reflects thus: ‘I am not the only one who is the owner of one’s kamma, the heir of one’s kamma; who has kamma as one’s origin, kamma as one’s relative, kamma as one’s resort; who will be the heir of whatever kamma, good or bad, that one does. All beings that come and go, that pass away and undergo rebirth, are owners of their kamma, heirs of their kamma; all have kamma as their origin, kamma as their relative, kamma as their resort; all will be heirs of whatever kamma, good or bad, that they do.’ As he often reflects on this theme, the path is generated. He pursues this path, develops it, and cultivates it. As he does so, the fetters are entirely abandoned and the underlying tendencies are uprooted.

If one clings to the above-mentioned views, one could abandon misconduct, and it could also lead to the uprooting of fetters. The same sutta includes other such skillful reflections or skillful views. However, as stated before, clinging to all views must eventually be abandoned.

A buddhist would want to attach to the noble eightfold path. It would be contradictory to claim that the eightfold path wouldn't encompass attachment. The distinction is whether you attach to the right things or not.

In the long run, the distinction between right/wrong may be abandonded, thanks to your initial attachment to right view. Dharma is merely a raft for crossing to the other side.

When would a Buddhist want to attach?

I suppose that people, Buddhists too, want all sorts of things at various times.

Perhaps you mean "When should they want".

The suttas are prescriptive (e.g. "you should X"), especially on topics of morality -- but they can also (perhaps e.g. on a topic like well-being or even nibbana) be seen as merely descriptive (e.g. "the consequence of X is Y") and let you decide for yourself (like a parent's saying, "if you do that, there will be consequences, so, choose wisely").

They welcome everything and don't clinging when it's gone.

I'm not sure "welcome" is the right word.

The suttas warn against "delighting" in what's pleasant. Perhaps instead a monk would be expected to "accept" or "note" rather than welcome -- I think that's what's meant by "guarding the senses" -- i.e. you sense the senses, you perceive perceptions, you even feel feelings, as anyone does, but -- remain detached, don't attach, don't delight in (nor recoil from ... and don't seek, but perhaps avoid).

I guess that ("not delighting") is because if a pleasant feeling goes when you weren't attached to it, that's OK, but if you were delighting in it then (per the doctrine of the 12 nidanas) that's what causes "craving" (and therefore "stress" and so on).

That doctrine might vary though -- e.g. perhaps it's what monks do and lay people don't (which is why lay people aren't apt to be monks) -- and as I said I think this is the doctrine of the suttas, perhaps other schools have another doctrine on the subject (e.g. perhaps about whether "avoiding" is a good practice).

But for the doctrine from the suttas, see e.g. nibbida of which e.g. here is a description: What is Nibbida?

They don't necessary cut existing attachments (knowledge, relationships)

Technically I'm not sure that those (e.g. relationships-between-people, and what's-been-learned-as-"knowledge") would be called "attachments" -- instead maybe they're called (identified as) sankharas -- Can anyone explain Sanskara / Sankara indepth?

The Pali word that's usually translated as "attachment" is upadana. That is most famously used in the compound word upādānakkhandha -- i.e. attaching or clinging to the (five) "aggregates" (i.e. khandha in Pali, or skandha in Sanskrit).

These five aggregates are "form", "sensation", etc. -- one of which is saṅkhāra (translated "mental formations"). But all the aggregates are sankharas, in the wider sense of sankhara -- Are all of the five aggregates saṅkhāras?

The five saṅkhāras are among (they're elements of) the twelve nidanas, and they're associated with (unwise) views of "self" (e.g. "I am form", or, "I am consciousness", or, "this feeling is mine", and so on).

Incidentally there's a difference between "aggregate" and "clinging-aggregate" -- Difference between aggregates and clinging-aggregates?

One of the similes (about "clinging") that I read was that if greed or craving or something is what motivates a thief to enter a room at night, then "clinging" or "attachment" is what happens when the thief puts their hand on something in the dark.

I think it's referenced in (or I recognised it in) Zen and the Art of Motorcycle Maintenance too, in the passage which includes the "South India Monkey Trap" -- small hole in a coconut with rice inside, monkey reaches in, makes a fist (to grab the rice) and is therefore trapped (because the fist is bigger than the hole), and to escape the trap the monkey must let go (and it asks, "What general advice...not specific advice...but what general advice would you give the poor monkey in circumstances like this?").

However, is there a case that a Buddhist wants to attach?

Maybe sort of. It's good to be kind to people, compassionate, to "put yourself in the place of another" and therefore not hurt them as you wouldn't want to be hurt. Maybe you shouldn't view yourself as "special".

After his enlightenment the Buddha consented to teach, though it might have been easier not to.

Lay life involves a lot of relationships -- see e.g. DN 31 -- though I wonder sometimes, if DN 31 is (or given that it is) Buddhist doctrine for Brahmins maybe that over-emphasises the Hindu notion of karma as social duty.

Religious life may depend on a relationship too, e.g. Kalyāṇa-mittatā, famously SN 45.2, perhaps also this topic.

The four brahmaviharas are described as the right social attitudes to have towards other people (compassion, kindness, respect/admiration, equanimity) -- but although these are "social" perhaps they're not meant to be "attachment".

Still that (i.e. thinking of others) might be better than thinking selfishly all the time -- if it is attachment maybe it's a better attachment. That brings up the topic of "generosity", incidentally, formally dāna -- possibly more than one purpose, e.g. to reduce attachment to your own "possessions", and (in places where there are Buddhists monks) it's because of dāna that monks can live -- it's a link between the lay and monastic societies.

There are many Zen stories about giving, by the way, e.g. The Giver Should Be Thankful, The Moon Cannot Be Stolen, The Thief Who Became a Disciple, Publishing the Sutras, and so on. The monks in the Pali suttas have no possessions to give, don't handle money, and so to that extent these suttas aren't very informative or detailed about giving and handling money and so on (though they do mention lay people giving monasteries, nuns receiving robes, and so on).