Spongebob: ESFJ (as stated before, he seems very focused on pleasing the customers and has often sacrificed doing or saying certain things to others in order to sustain peace. Note how he always goes to work every day, freaks out when he's late (then becomes apologetic and seems to enjoy doing things by the book, he never gets bored of doing the same thing every day as long as it pleases those around him. The Ne that he does express could just be a well developed tertiary function which ESFJs have.)
Squidward: Unhealthy ISFP (I don't see him as being an Si+Te user at all, he doesn't seem to care much about any of his responsibilities, especially if he doesn't enjoy them. He's pretty shitty at his job and doesn't handle responsibility very well. And remember how sick he got of routine when he ended up moving to that squidwill or whatever place? He puts a lot of value in physical aesthetics (I've noticed Fi-Se users to do this often) and puts most of his effort in the things he likes (practices this clarinet without caring whether others it him or not). Perhaps since he's unhealthy, he can portray ENTJ since that's his shadow which may be why people pin him as an INTJ?)
Patrick: ISFP (Patrick is pretty stupid and that makes it difficult to pin point his type. But he seems to love to follow spongebob on adventures and seems like a very flexible character. I don't see him as being an extrovert though.)
Mrs. Puff: ISFJ (I don't think she'd be able to put up with Spongebob as well if she wasn't and demonstrate so much patience towards him despite not liking him. The way she teaches boating school and how (rather than improvising in order to get spongebob to pass or changing her method) she goes about it the traditional way while generally suppressing his own emotions to sustain peace).
Mr. Crabs: ESTJ (I don't see him as being a Fe user, I think his method of going about to get what he wants seems quite Te+Fi, but I may be wrong. He seems to pinpoint and remember a lot of details about objects, where he got them from, etc (like in that episode where spongebob thought he was a robot and wanted to fuck all his stuff up) so this reason I'd lean towards Si user.)
Pearl: ESFP (For the most part, I think he personality is pretty constant compared to the other characters, and that's probably since she's not aired often. Her focus seems very Se+Fi (especially at her prom and such). Don't see much if any Ne in her. And I think ESFP is what most people guessed anyways, so there's probably little arguments on that).
Plankton: INTJ or messed up IxFP (Absolutely no use of Ne (or if any, it's a very underdeveloped function in him). But I also question the Ni+Te which other people claims he has. He doesn't really spend all that much time thinking his ideas through, considering whether they're logical and checking whether they're fool proof. He just kind of... wings it with similar methods every time and gets fucked over. But that's probably since this is a kids show and/or he's kind of stupid. On that note, I don't see him as being an Ti+Fe user either. But why not an unhealthy Fi+Te user, who's holding onto his messed up values due to past events between the two and using his under developed Te to obtain them? That might imply that he's a fucked up INFP or ISFP, and I doubt that. So I'd go INTJ I guess

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Sandy: ESTP (I can see why some might argue ENTP or ENFP for her, but I think she's a fairly clear Ti+Fe user. Her love for extreme sports and physical activity as well as competition (note how much she loves the karate match her and spongebob had that one time) makes me lean towards Se. I've seen little to none examples of her demonstrating NE and ESTP can also have a fascination and interest with science.)
Gary: INTP (I'm basing this off of that one dream episode where Gary actually spoke.)
The Flying Dutchman: ENTP (because he's fucking awesome.)
(not sure whether I should feel proud of ashamed that I took half an hour to analyze a cartoon show).