How do these 3 ways of saying “It can't be helped” differ?

Question

So I have heard:

• 仕方がない
• しょうがない
• 仕方あるまい

And the translation I've seen for all 3 is: "It can't be helped." Is there a particular difference among the 3?

Edit: Secondary question. Why isn't 仕方あるまい, 仕方があるまい?

Answer 1

仕方がない

仕方：　し is the conjunctive form of する (to do) and 仕 is 当て字。方 means way or method.
仕方がない：(lit.) する手段・方法がない There is no way to do a thing.
From this idea comes these idiomatic meanings:

1) やむを得ない - Unavoidable, inevitable. (Hence, It can't be helped. Nothing can be done about it.)
2) どうにもならない - A thing for which it is not possible to do anything about. No matter what one does, it is not possible to make it work. (Thus this definition of 仕方ない is also rendered "It can't be helped" in English)
3) はなはだしく悪い、改めようがない(as in 仕方ない奴) - A person that nothing can be done about (context determines severity, often used relatively lightly.)

仕様がない＝しようがない＝しょうがない.

Breaking 仕様 down:
In the same way as for 仕方がない、し is the conjunctive form of する (to do) and 仕 is 当て字。 Originally しよう, people pronounced it しょう enough that it became acceptable to write it that way too (as a matter of notation, not spoken word vs written word).
仕様： the way a thing is done. The method by which an action is taken.
仕様がない： (lit.) There is no way/method by which action can be taken.

Thus we get these idiomatic meanings:

1) なすすべがない - There is no means by which to accomplish a thing
2) よい方法がない - There is no worthwhile/plausible way/method in which to do a thing
他に良い手段がない、やむを得ない - There is no other way, inevitable
3) あきれるほどひどい、手に負えない- Be uncontrollable; be incorrigible; be beyond one's control; defying authority (from the idea that "Nothing can be done about such a person")

So the answer to your question for these two is in the difference in the idiomatic meanings derived from these expressions. Judging by my literal translations of the definitions, they overlap quite a bit, so much that it might be considered futile to try and separate them. It requires a thorough understanding of each of the idiomatic expressions used to define 仕方がない and 仕様がない to really see and feel the difference. However, it is my opinion that in practice one would be hard pressed to identify the nuances of these expressions in the usages of 仕方がない and 仕様がない。

As for 仕方あるまい, technically it is 仕方があるまい, just like 仕方ない is 仕方がない. It is simply an abbreviation of the particle. So the difference from the others boils down to ない and あるまい which is the difference between declaring "there is none" and stating with less certainty "there is most likely/probably none" (simplified translation, I don't think there is a precise way to render だろう・まい in English)

This information was compiled from the 広辞苑(こうじえん)、日本国語大辞典(にっぽんこくごだいじてん)、and 大辞林(だいじりん) dictionaries.

Answer 2

仕方がない should be fairly evident meaning-wise. しょうがない is a variation of 仕様がない, and 仕様 effectively has the same meaning as 仕方, so they are virtually identical in meaning, however しょうがない is more used in spoken language while 仕方がない can be used formally.

仕方あるまい is the same as well, except it uses the あるまい construction, which is essentially a more formalized way of saying it. It could also mean ないだろう, but basically you're looking at three phrases that mean the same thing.