Shat is the widely accepted past-tense inflection of shit. It was formed by analogy with the verb sit, which becomes sat in the past tense. It was originally a humorous and slightly sanitized version of the curse word, but it has become the standard form.
There are two alternatives: (1) shitted, which appears about once for every instance of shat; and (2) shit, uninflected in the past tense. The second option likens the word to verbs like fit and knit, which are usually uninflected in most varieties of English.
No usage guides that we know of cover this word, so it’s best to go along with conventional, real-world usage. Right now, the conventional usage favors shat. Here are a few examples form recent sources:
In many ways a one-man Metalocalypse, Putnam ate, puked, and shat metal. [AV Club]
Were we when (as one commenter on our theatre blog reported) some guy dropped his pants in a recent performance piece and shat on stage? [The Guardian]
[T]hey, in turn, have started to forget that it’s their right to object when shat upon. [New Yorker]
But the other inflections do appear occasionally—for example:
If his music sucked no one would care that he shit on stage and beat people up 20 years ago. [transcribed in Oregon Music News]
He earned that right by getting shitted on right after he was drafted and constantly was in trade talks by stupid fans. [Hoops Vibe]