Shawn - ENTP - He's obviously an E-type, and likely N--He makes the observations and uses intuition to get to the truth. I think he's probably T because he doesn't do much worrying about how he makes people feel, and P because of his general laid-back demeanor.
Gus - ISTJ - As a foil character, it makes sense that Gus would mostly have opposite traits to Shawn. Again, though, he's logical and not overly emotional (I think? It's been a while since I've seen the show). But he's very much a perfectionist, an introvert, and far more conventional than Shawn.
Lassie - ISTJ - He's also meant to sharply contrast Shawn, and he's not emotional. Yeah.
Jules - E?F? - As the romantic interest to Shawn, well.. Um.. Like I said, it's been awhile since I've seen the show.
Shawn's dad (I'm forgetting his name at the moment) - ENTJ. Perhaps?
Eh, perhaps. Although he could just have both hyper-developed Se and Ne. Many of the connections he makes are very intuitive, and I think it would be hard for a sensor to make them so quickly and accurately.
Shawn - ENTP 7w6 with overdeveloped Se.
Gus - ISFJ with good Ti. The man has no Te whatsoever and can be a major pushover. also, definitely a 6w7- got a lot of nervous energy.
Lassie- ISTJ, probably 1w9 or 6w5.
Shawn's dad - ExTJ
Shawn: ENTP w/ Se. He is so obviously Ne Dom, though. He may be great with sensing what's around him, but when it comes down to it, his ability to recognize patterns and analyze data make him an obvious Ne.
Gus: ISFJ. This makes sense, as he is a foil to Shawn.
Juliet: Always thought of her as an ISTJ. She has those feeling moments, but I imagine this is just poking fun at her being the only major female character in the show. For the most part, her decisions seem to reflect Te, and she never gets pushed around. Si, because she seems very conservative/traditional; rarely questions the norm.
Lassie: ESTJ, to the absolute extreme. I don't think I really need to defend my reasoning here.
Henry: (Why is the dad always named Henry? Nvm) ENTJ. Very strong Te here, with plenty of Ni to back it up, and it would make sense that he and Shawn have shadow functions.
Chief Vick: I'm gonna go with ESTJ. She has the bossiness of a Te dom, but rarely does she question the way things are done.
Since the show pretty much revolves around Shawn, it makes sense that most of the characters are SJs, since they really juxtapose his crazy Ne.
Shawn - ENTP overall. I do see where the Se comes in.
Gus - At first I thought ISTJ, but ISFJ does seem to better fit him. I see more Fe than Te.
Jules - This is a tough one, but I can see ISFJ for her too or maybe INFJ. I can't help but see Fe-Ti as her 2nd and 3rd functions.
Lassie - He strikes more as an ISTJ, but I can see ESTJ as well.
Chief Vic - I can see ESTJ or her, but this one is tough. Perhaps she's not as well developed as the rest of the crew.
Shawn's Dad - Either INTJ or ENTJ - leaning more INTJ. I'm leaning more of a Ni lead. I see see the father-son combo connect on having N-leads with the father as Ni-lead and the son as Ne-lead.