Could have and should have are verbs, and verb tenses indicate when an action took place.
These helping verbs are combined with other verbs to express something that was possible in the past but didn't actually happen, as in: "We could have walked to the park, but we stayed home instead." Technically speaking, could have and should have are modal verbs in the past tense, and they precede the past participle of another verb: I should have known, she could have reminded me, we would have remembered.
These sentences get it right by using the verbs should have and could have:
Some say the prime minister has been lax and should have moved earlier and more decisively. (Washington Times)
Who could have predicted that everything was about to change? (The Guardian)
Could have and should have are often abbreviated as the contractions could've and should've. When you say could've and should've aloud, they sound a lot like could of and should of. That's why many people mistakenly use could of and should of. Don't feel bad if you sometimes get these modal verbs wrong: people have been making this error in writing for over 200 years!
The authors of these sentence should have known better:
Bauer was also asked if his actions Sunday in Kansas City could of had a negative impact on the team. (Los Angeles Times)
They fell for the old "home prices can only go up" line and borrowed way more than they ever should of. (Inc.)
Just remember that have is a verb, while of is a preposition. You need a verb in the verb phrases could have and should have.