I won’t and I’ll not are both short forms of I will not. Both are used in English.
Are there any situations where one is preferred over other?
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The following is from my own (and, I trust, a general British) perspective. This one would be very difficult to provide supporting references for.
I won't is the choice when an emphasised statement or retort is given:
"You must do two hours unpaid overtime tonight."
"I won't! / I will not!" (not "I'll not!"; won't here connotes strong will)
"There's no other way – you'll have to drive the truck through the minefield."
"I won't stand a chance!" (preferred) ("I'll not stand a chance!" sounds weaker)
"I won't inform the police that the man who ran on to the pitch and thumped the player was your brother." (won't stresses my – perhaps concessive – non-involvement; I'll not would be an unemphasised statement of intent)
I'll not is also short for I shall not, but the will / shall complexities seem to be very unresolved and argued over.
Normally, you will use I won't. Only when not is somehow strongly linked to a phrase would you perhaps sometimes leave it as such and simply use I'll rather than abbreviate not. But it is fairly rare. I believe you will find it more frequently in older English. Using I won't always is your safest bet.