Native English speakers tend to use all these auxiliary/modal verbs without giving a thought to which is most appropriate. The choice depends largely on preference and on the context.
May is generally associated with seeking or giving authority/permission and with uncertainty.
May I say something?
Yes, you certainly may.
Can is associated more with ability/skills.
Can you finish the job today?
I may not be able to.
Could and might are both used mainly after a main verb in the past tense and before conditions. So:
He says he may go this weekend
He said he might go this weekend.
And we often say things like:
I could do it if I had the tools
I might go to the beach if the sun shines.
So you may be able to work abroad and you might be able to work abroad mean exactly the same thing but the latter works better after a main verb in the past tense.
You can work abroad states that you have either the skills or the go-ahead (or both) to do so.
You could work abroad would generally (but not always) be used before a condition although a speaker might well say this in response to a question.
I can't find work in this country.
Well, you could work abroad.
However, there are whole books on the usage of these verbs and you will find guidance at numerous sites including these: