Should Have or Should Of?

The Quick Answer
Should've sounds like should of, but it is short for should have. (This is the same for could've and would've.)

Should Have or Should Of?

The contraction should've can only be expanded to should have. The incorrect expansion should of has developed from how should've is pronounced. This is also true for would have and could have. For example:
  • I was married by a judge. I should have asked for a jury. (Groucho Marx)
  • I have offended God and mankind because my work didn't reach the quality it should have. (Leonardo da Vinci)
  • Had I become a priest, the sermons would of been electric!

See Also

What are contractions? Using apostrophes More on abbreviations