When do I use "may", and when do I use "might"?

I use them interchangeably.

Formally there is a reason to use “may” rather than “can” when asking for permisslon.

“Can I do that” implies ability or physical prowess. “Can I leap over that building in a single bound”?

“May I leap over that building” is asking for permission.

“Might I leap over that building” implies that I am a nineteenth century British pedant.

^^ “Might I leap over that building” makes me think the speaker is contemplating the act.

Might only has one common definition: a verb to express possibility.
May, on the other hand, has two different meanings. From dictionary.com, here are the two definitions:
1. verb used to express possibility: “It may rain.”
2. verb used to express opportunity or permission: “You may enter.”
Like other words, the meaning you use is based on context.

Here are some examples I came up with:

You might have to do this. vs. You may have to do this.
In this case, they mean virtually the same thing: there’s a chance you’ll have to do this.

You might do this. vs. You may do this.
Now, there’s a bigger difference between the two sentences. The former means there’s a chance you’ll do this. The latter is ambiguous: It could mean (def. 1) There’s a chance you’ll do this, or (def. 2) You have permission to do this.

Again, the only real way to know which idea the sentence is trying to convey is by context.

Okay – I understand that “may” is used to ask for/grant permission. I thought there were rules on when to use “may” vs “might” to talk about possibilities – but it looks like I was wrong there. Thank you, I’m happy to learn things are easier than I thought! :)

It may never be possible to sort it all out.

It might never be possible to sort it all out

Mightn’t it?

Might makes right.

@longgone More or less, if you’re trying to indicate a possibility, they’re interchangeable. For now, why not play it safe and just use might for possibilites and may for permission?

Technically “might” is the past tense of “may”, so when saying something “he may have left already”, it should be “he might have left already”. It’s become optional for the most part now. Sometimes people use “might” to avoid confusion with the “permission” meaning of “may”. “We may not see him” can be “we don’t have permission to see him”, so people use “might” to indicate “there’s little chance we will see him.”

Another rule I read is that “may” indicates a larger possibility than “might”. “We may get there on time” vs. “we might get there on time”—the latter is supposed to indicate less of a chance.

The little jingle used by children when they see the first star to appear at night:

Star light, star bright,
The first star I see tonight;
I wish I may, I wish I might,
Have the wish I wish tonight.

GQ. I never thought about this particular little bit of English grammar before.

To be accurate or a purist, it’s usage rather than grammar.

http://grammarist.com/usage/grammar-usage/

@DominicY I like your first paragraph, it seems “right” to me.

@gailcalled While adding tags, I thought about how the issue wasn’t really grammar-related. “Usage”, remained on the tip of my tongue. I had planned to defend myself by saying that most jellies interested in my question are likely to have “grammar” as one of the topics they’re following. So…that’s my excuse, and thanks for the link!

She probably was correcting me.

I may help you in that, however I might not be helpful at all.

Well, the ngram viewer suggests that it’s far more common to use “might” these days, which fits with what my spider sense tells me.

“May” also seems rather more formal to me, which may, perhaps, explain its relative rarity compared to might.

(Idiomatically it’s I may help you with or through that.)

@gailcalled – In my experience, the “rules” are no such thing.

Language is driven by the communication it embodies; while I certainly remarked on the unusual usage, it didn’t strike me as breaking any “rule” of language. In fact, I found the expression rather charming.

Dependent prepositions are remarkably flexible.

(I said “idiomatically” and did not mention any rule.)

That’s a get out.

(“Get out” is an expression I am not familiar with.)

a get out of jail free card.

I can’t parse “idiomatically” without reading you as appealing to some kind of rule of usage. There are none. It’s as simple as that.

Some people say that “may” indicates something that is more likely than “might”. Also you have to use “might” when you’re talking about something in the past. And if you’re taking about something not happening you should use “might”.

http://www.quickanddirtytips.com/education/grammar/may-versus-might?page=all

I do not find these people’s arguments convincing. The idea that you have to say “Squiggly might have gone to the party” instead of “Squiggly may have gone to the party” is silly.