Urahara: ENTP
He is too flamboyant to be an INTJ, and he isn't really very righteous. Also, I can't really see efficiency (Te) in him, he just doesn't come across as a product-oriented type, but rather, has a decently strong Ti. He seemed more amused and extroverted than a typical INTP, and also, he seemed to view everything as fun, seeing how he has a small smile on his face all the time. Besides that, he had a strong use of Ne when it showed that he even prepared a portal for Ichigo to rescue Orihime, and his movements, while being calculated, aren't actually very righteous/ethical motive, but more of a cause for amusement and to help out others.
Ichigo: XXXX never liked him, never will.
Rukia: ISTX/ INTJ
I originally thought ISTJ, since she seemed to carefully follow the rules and take after Byakuya, but her dominant function doesn't really seemed to be Si.
Toshiro: IXTJ
I can't tell. Sometimes, he surprises everyone by not upholding the rules, other times, he quietly uphold and submit. Probably INTJ, but he had a strong Te, just that it's not dominant.
Orihime: ESFJ.
She can burn herself, she's literally just there to cry, pretend to be innocent, and be a perfect character for fan service.
Sado: INFP
He is extremely loyal (Dominant Fi). Doesn't seemed to use Se very much, so I'll guess INFP.
Starrk: INTP
Ulquiorra: INTP
He follows orders, but he's dominant function seemed to be Ti over Ni.
Shunsui: ENTP.
...I don't even need to explain, seriously, just look at him.
Shinji: XXTP
I can't be sure about Shinji.
Gin: XNTP
Aizen: INFJ
Byakuya: ISTJ
Like what
@Eska said, I wouldn't type him as an ESTJ since he doesn't impose anything on others. He punished them for breaking the rules because he is /obliged/ to. Also, it's clear that he likes to be alone, and he doesn't really like company, as seen from how awkward he gets around others.
Ishida: INTJ