"I won't" vs. "I'll not"

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?

Solution 1:

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.

Solution 2:

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.