style - How do you write a character's internal monologue?

I've seen two main methods of writing a character's inner monologue

One is to use italics
Dean is being silly thought Sam

Another is to quote the text
"Dean is being silly" thought Sam

Or you could avoid it altogether and simply relate what the character thought
Sam thought Dean was being silly.

Personally I don't like to quote thoughts as it can lead to confusion between their speech.

What is the correct way to indicate that you're quoting their internal monologue as opposed to your characters' spoken word?

Answer

I use italics, I find it is the clearest way to define thought as different to speech, and denoting actual thought as a form of dialogue can help draw distinctions between actual thought and narrative.

In third person narrative it is common to write from the perspective of the character in question, and colour the tone of your writing with the way they think. As an example I'd refer you to the First Law trilogy by Joe Abercrombie. Each chapter is devoted to the perspective of a particular character, and each is written in third person limited perspective (so you're effectively inside the character's head.)

The inquisitor is prone to self-analysis, so even though the narrative is effectively coloured with his way of thinking he also thinks as dialogue — in italics — as a way of differentiating those thoughts. Conversely the barbarian Nine Fingers basically never thinks, his narrative is more straight forward and inner dialogue is rarely — if ever — used.

Most of the authors I read do the same so it's certainly a strong convention, but as PraveshParekh says, all are viable options. So while there is no “correct” way, if you are going to use inner dialogue then I believe that sticking to italics — and convention — helps avoid confusion.