To decline the supine, use the fourth declension. Since you only need the accusative and ablative singular, the only endings you use are -um and -u. The tricky part, if you don't have a dictionary, is figuring out the stem to which to add the -um (used for purpose with verbs of motion; translated like an infinitive) or -u (used with adjectives and sometimes verbs). This depends on the conjugation to which the verb belongs. If you have a dictionary, the fourth principal part is usually the supine, but with a us ending.
To form the supine for verbs in the:
- 1st conjugation - remove ending but keep thematic vowel (a); add a -t- and then add the 4th declension endings (either -um or -u), as in paratum.
- 2nd conjugation - remove ending and thematic vowel (e); add -it- and then add the 4th declension endings (either -um or -u), as in habitum
- 3rd conjugation, you must deal with irregularities. The supine of mittere is missum, but the supine of capere is captum. Although this may not be entirely predictable, the supine is usually used as the fourth principal part in the dictionary entry for the verb, so it should be familiar.
- 4th conjugation - remove ending but keep thematic vowel (i); add a -t- and then add the 4th declension endings (either -um or -u), as in auditum.
- Venerunt visum they came to see.
- Mirabile dictu wondrous to say.