This is an easy one with an interesting explanation.
Would have is correct.
Would of is not correct.
However, “would of” has crept into the language because when “would have” is abbreviated to “would’ve”, and spoken out loud, it sounds like “would of”.
“She would have gone out, but she didn’t have anything to wear.”
“She would’ve gone out, but she didn’t have anything to wear.”
This works throughout the range of words – “might have”, “would have”, “should have”, etc.
For more troublesome pairs, choose the category from the cloud over to your right, or click here.
LibroEditing proofreading, editing, transcription, localisation 
Jen
June 3, 2011 at 7:37 am
Thank you, this is one which really grates on me when I see it written down!
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Liz at Libro
June 4, 2011 at 5:43 am
Thanks Jen – a few people find that!
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Luise Voigt
July 4, 2011 at 4:44 pm
Really clear website , regards for this post.
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