Formation of supinum is pretty complicated. Most handbooks advise to memorize this form of every verb, but it's not an absolute necessity.

I conjugation : -are > -atum.
Exceptional forms occur only when perfectum is in -ui or in -i with vowel lengthening :
iuvare > iutum, vetare > vetitum, sonare > sonitum.

II conjugation :
if perfectum is in -ui, then -itum (habere, habeo, habui, habitum)
Exceptions: tenere > tentum (retinere > retentum, sustinere > sustentum), docere > doctum, miscere > mixtum, censere > censum.
If perfectum is in -vi : change of -vi into -tum : movi > motum, flevi > fletum, implevi > impletum
except : solvi > solutum, cavi > cautum.
If perfectum is in not in -ui or -evi :
-si > -sum : iussi > iussum, mansi > mansum, arsi > arsum, risi > risum
-xi > -ctum : auxi > auctum
-di > -sum or -ssum : videre (vidi) > visum , respondere (respondi) > responsum, sedere (sedi) > sessum.

III conjugation:
If it's -uere : -uere > -utum : constitutum.
Forms with perfectum -ui change -ui into -itum :
alere, alui, alitum, ponere, posui, positum, gignere, genui, genitum
but some verbs lose -i-: deserere > desertum, consulere > consultum, rapere > raptum, colere > cultum, eripere > ereptum

When perfectum is in -si, -xi:
-xi > -ctum (dictum, ductum, victum, tractum, cinctum, surrectum, respectum), exception: finxi > finctum, fluxi > fluxum
-psi > -ptum (sumptum, scriptum)
-si > -ssum in verbs in mittere : missum
-si > -sum (clausum, divisum, sparsum, laesum)
-ssi > -ssum (pressum, cessum), but : gerere / gessi / gestum

When perfectum has other consonants or i before -i:
Some verbs which only change vowels (a to e) undo this change : agere > actum, facere > factum, capere > captum, cogere (coegi) > coactum, frangere (fregi) > fractum, iacere > iactum ; exigere (exegi) > exactum, contingere (contegi) > contactum
-gi, -ci, -qui > -ctum : vici > victum, reliqui > relictum, legi > lectum, rupi > ruptum, verbs in -ficere: efficere > effectum, similarly : deficere, interficere, conficere, afficere
exception: fugi > fugitum
-pi > -ptum (accipere, accepi, acceptum; same: recipere, excipere, incipere, suscipere, praecipere)
-ii > -itum (desinere)
-vi > -tum (petitum, quaesitum, cretum, cupitum), exception : cognovi > cognitum
-di > -sum (fusum, effusum, reversum, ostensum, defensum, occisum, descensum), but intendi > intentum

Verbs with reduplication lose it :
verbs in -didi > -ditum : traditum, editum
The other verbs :
cadere > casum, caedere > caesum, canere > cantum, parcere > parsum, fallere > falsum, tendere > tentum, tangere > tactum, currere > cursum, pellere > pulsum, pendere > pensum, parere > partum.

IV conjugation: from perfectum:
-ivi > -itum
-ni > -ntum, -ui > -tum, -si > -sum
Reduplication : repperi > repertum.

Not all irregular verbs have a supinum form. Verbs in -ire, -ferre have it:
ire > itum : ire > itum, with prefixes : exire > exitum and so on
ferre > latum : ferre > latum, deferre > delatum, referre > relatum
with change of prefix : afferre > allatum, auferre > ablatum, conferre > collatum, differre > dilatum, inferre > illatum, offerre > oblatum, tollere > sublatum
esse does not have supinum, but has a form : futurum.
Deponentia - here we show the form of passive participle in -us. Deponentia are not very numerous and often have irregular forms here.
-ari > -atus
-eri > -itus in vereri, but see others below:
-qui > -cutus : sequi > secutus, loqui > locutus
pati > passus, fateri > fassus, uti > usus
-gredi > -gressus (ingredi, egredi)
queri > questus, oriri > ortus, experiri > expertus, reri > ratus, mori > mortuus
-sci > -tus : nasci > natus, irasci > iratus
proficisci > profectus, confiteri > confessus