We often call Shia Labeouf the King of the Small-Time Criminals . . . because he gets arrested for petty things like smoking on a public sidewalk where lighting up is banned and loitering drunkenly in a Walgreen's.

But it turns out Shia's dad is a BIG-TIME criminal.   Jeffrey Craig LaBeouf did time from 1981 to 1983 due to ATTEMPTED RAPE.   And because of that, he's a registered sex offender to this day. (--Here's his mugshot.) (Radar Online)

According to RadarOnline.com, Jeffrey faced additional legal troubles around that time.   In apparently separate cases, he was charged with kidnapping and assault with a deadly weapon.

But Radar says, quote, "the legal outcomes of these charges are unknown."

More recently, Jeffrey was accused of sexual harassment on the set of Shia's first major gig, the Disney Channel show "Even Stevens".   He also once attacked a gay Disney exec who tried to give him a congratulatory hug.

Shia has claimed in interviews that his father was physically and verbally abusive . . . and once held a gun to his head during a Vietnam flashback.