Often told that supine is used for Verbs of motion while 'ut/ne' for other verbs. An explanation here could help more.
The conjunction ut/ne is a general way to express a purpose (also called the ut finale, because it is only one of the uses of ut). The supine in -um is a very specialized form used only to express the purpose of a movement.
The supine is used with all kinds of verbs of motion:
- Leones spectatum venimus. We came to see the lions.
- Romam iit auxilium rogatum. He went to Rome to ask for help.
- Post solem occasum cubitum discesserunt. After sunset they left to rest.
You will also find verbs that do not actually describe someone moving, but someone causing motion. The supine then still signifies the purpose of the caused motion:
- Caesar milites frumentatum misit. Caesar sent soldiers to forage.
- Hannibal patriam defensum revocatus est. Hannibal was called back to defend his home country.
You cannot use the supine with verbs that do not fall in this category:
- Imperator Cypri mansit ut classem compararet. The general remained in Cyprus to assemble a fleet. No supine possible here.
And even with verbs of motion, you do not have to use the supine; in fact, it is not particularly common:
- Carthaginem est profectus ut patriam defenderet. He departed for Carthage to defend his fatherland.
Which is just as well, because the supine is not very flexible. It allows no simple negation like ut ne (“so that not, lest”) and no passive. A nice illustration of a sentence which first uses the supine, then switches to ut, is this timeless observation of human behaviour by Ovid (Ars amatoria 1,99):
- Spectatum veniunt, veniunt, spectentur ut ipsae. The come to see, they come to be seen.
Aside from all that – though technically still a use with a verb of motion – the supine is used with iri (which is the passive infinitive of ire) to form the future passive infinitive, e.g.:
- Puto inimicos expugnatum iri. I believe the enemies will be defeated.
This should be intuitive to English speakers, who use “going to” in quite a similar way. But this is also quite a rare construction.