助ける is the most common way to write たすける using kanji. It means to save someone from danger, the same as expressed in the word 救助する. It's the only kanji with たすける as a joyo reading.
扶ける means to lend assistance or help someone (力を貸す). But this meaning can also be written as 助ける, thanks to the fact that たすける is not a joyo reading for this kanji.
輔ける means to advise or assist someone, as in 補佐・輔佐する. To reiterate, it's not a joyo reading.
My dictionary doesn't have explanations for the other ways of writing it you mentioned, and my kanji dictionary lists all of these except for 助 as non-joyo readings. Here is the dictionary entry with the information I mentioned for the 3 renderings above.
The instances in BCCWJ for each (or related forms) are
助ける - 6708
扶ける - 21
救ける - 42
援ける - 35
輔ける - 6
佐ける - 2
佑ける - 0
In short, 助ける is the primary way of writing it, and you will occasionally come across other renderings, the meanings of which can mostly be guessed from their individual kanji nuances.