Amphiboly
Amphiboly is when a sentence can be read in two different ways because of its structure—e.g. unclear attachment of a phrase or ambiguous reference—and the argument uses the confusion. Grammar or word order makes the sentence ambiguous. The fallacy is to rely on one reading to draw a conclusion when the other reading would not support it, or to exploit the ambiguity to mislead. Fixing the grammar or rephrasing usually removes the ambiguity.
Examples
The police shot the man with a gun.
I saw the man on the hill with the telescope.
The council will discuss rubbish in the meeting.
She told her daughter she was leaving.
The dog chased the cat that was eating the bird in the garden.