I do not have anything formal on these, however I will comment on my personal observations. Because these words are all close synonyms, I do not feel there is a substantive difference in their meanings. Thus their difference comes down more to feeling and soft differences. It is a bit like trying to explain the differences between "gigantic," "ginormous," and "humongous."
This is not necessarily meant to be a full and robust answer, but my thoughts are too long for a comment.
Let me first address 곱다. I feel that this word is less commonly used to express physical beauty than the other three words. It connotes a feeling of softness, of genuine internal benevolence. One common translation I have seen is "fair," as in "a fair virgin." It captures not just a feeling of physical attractiveness, but moreover a sense of beneficence. 곱다 is more commonly applied to beauty of overall character, and possesses a literary feel.
Now for the other three words. These form a rather close synonym group. I honestly believe that there is not going to be a "right" answer to the differences in their meanings. Different people will use them differently.
아름답다 feels a little more formal than 예쁘다 and 이쁘다. The gap in formality is small though. Just in my experience, 아름답다 is a word less used by children than 예쁘다/이쁘다. (But this is not a hard and fast rule of course. I do not have any percentages of use between the two). 아름답다 at times carries a feeling of more distance between the thing being called beautiful.
One instance where 아름답다 feels more appropriate than 예쁘다/이쁘다 would be something like talking of Queen Elizabeth II of England.
엘리자베스 여왕은 너무 아름답다. Queen Elizabeth is very beautiful.
Most of us do not see 80-year old women as the paragon of physical and sensuous beauty. But Queen Elizabeth can still be considered "beautiful." And there is a distance or elevation between us commoners and Her Majesty. This is one instance where there could be some overlap into the usage of 곱다.
One last thing I will mention about 아름답다 is that it can be used to describe nature, whereas I feel it would be less common to see something like
경치가 예쁘다 (The view is "beautiful.")
This is not to say that some Korean out there would not say it, it just would be less fitting I feel.
It is quite natural however to say something like
Now for 예쁘다/이쁘다. These two words are (in my opinion) almost the exact same word. They may even be rooted in the same word, with two variants in pronunciation being later establish orthographically. Nevertheless, whatever their origin, I believe the two words are one-for-one interchangeable. (Listen to your emotions on this one. Honestly).
Most uses of 예쁘다/이쁘다 will revolve around physical beauty of some sort. It can be used to describe people and their various features (face, voice, eyes, smile, etc.) 예쁘다/이쁘다 can also be used to say something has gone beautifully or has been agreeable:
애들이 조용히 놀아서 정말 예쁘다! ( The children played so quietly--it was quite lovely!)
As a final note, it would sound awkward to me to use any of these words when describing the physical features of a man. I am sure that there are cases where they are used, but I do not see most Korean men being generally accepting of having the verb 예쁘다 associated with one of their features. Words like 멋지다 or 잘 생기다 would be better choices.