MIGHT HAVE vs COULD HAVE

We use might have to say

that something is possible or something happened

• Polly's very late. She may have missed her train. (= It is possible that she missed ...)

that something was possible but did not happen

• You were stupid to try climbing up there. You might have killed yourself. (but you didn't)

Examples:

• If she hadn't been so bad-tempered, I might have married her. (=but I didn't)
• I'm so sorry. We might have done something to help you. (=but we didn't) 
• ...Therefore, meningitis patients with negative blood cultures might have been missed. (=there is a possibility that these patients were not recognised) 
• You are so lucky because it's only a dent. It might have been worse! (= but it wasn't that bad)
• All that Darwin did was to suggest a plausible explanation of how it might have occurred. (=perhaps that's how it happened)
• More significant is the notion that Muhammad's death might have been a result of food poisoning.(=perhaps he died of food poisoning)
• It’s ten o’clock. They might have arrived now. (=perhaps they arrived)
• He might have easily failed that test! (=but he passed it)

Could have & Might have

Could have & might have can be used interchangeably when we want to make a guess about something - it's possible / probable that something happened.

John is late.

• He could have / might have got stuck in traffic.
• He could have / might have forgotten that we were meeting today.
• He could have / might have overslept.

However, if we want to express the ability, we can't use might.

Compare:

• If she hadn't died in that accident, I might have married her. (=the reason I didn't marry her has nothing to do with my willingness, but with possibility ) 
• I could have married anybody I wanted to. (=I was able to choose anyone I liked, but I didn't want to)

• You were stupid to try climbing up there. You might have killed yourself. (It wasn't your intention to kill yourself - there was a possibility for that to happen in a dangerous situation)
• I was so angry I could have killed her! (Luckily, I decided to put the gun down and calm yourself. We aren't talking about possibility here, but about your willingness to try or not to try to do something) 

• If you had arrived early for the match, you might have bought the ticket. (there was a possibility, maybe you would've bought the ticket, maybe not)
• If you had arrived early for the match you could have bought the ticket. (you would have been able to do something - we don't question the possibility for something to happen)
• If you had arrived early for the match, you would have bought the ticket. (we want to express certainty about the activity)

Might have and Could have can also be used to show that we are annoyed with something or someone:

• You might have cleaned up before you left.
• You could have helped me - why did you just sit and watch?
• You could have at least tried!