Which of the following statements are correct regarding State Legislatures?
A. Notwithstanding anything in part XVII, but subject to the provisions of Article 348, business in the Legislature of a State shall be transacted in the official language or languages of the State or in Hindi or in English.
B. The Speaker of the Legislative Assembly or Chairman of the Legislative Council can permit any member to speak in his mother tongue.
C. Only the Supreme Court is entitled to inquire into the proceeding of the State Legislature.

To determine the correct statements regarding the State Legislatures in India, let us analyze each statement individually based on the provisions of the Constitution of India:

Statement A: "Notwithstanding anything in part XVII, but subject to the provisions of Article 348, business in the Legislature of a State shall be transacted in the official language or languages of the State or in Hindi or in English."
According to Article 210 of the Constitution of India, business in the Legislature of a State shall be transacted in the official language or languages of the State, or in Hindi, or in English. This is subject to Article 348 (which relates to the language of the Supreme Court and High Courts). Therefore, Statement A is correct.

Statement B: "The Speaker of the Legislative Assembly or Chairman of the Legislative Council can permit any member to speak in his mother tongue."
According to the proviso to Article 210(1), if a member cannot adequately express themselves in any of the languages specified (official languages of the State, Hindi, or English), the Speaker of the Legislative Assembly or the Chairman of the Legislative Council (or the person acting as such) may permit that member to address the House in their mother tongue. Therefore, Statement B is correct.

Statement C: "Only the Supreme Court is entitled to inquire into the proceeding of the State Legislature."
According to Article 212 of the Constitution of India, the validity of any proceedings in the Legislature of a State cannot be called into question in any court on the ground of any alleged irregularity of procedure. This restriction applies to all courts, including the Supreme Court and High Courts. Thus, the statement that "Only the Supreme Court is entitled to inquire..." is incorrect. The courts generally do not inquire into the internal proceedings of the State Legislature. Therefore, Statement C is incorrect.

Since statements A and B are correct and statement C is incorrect, the correct option is A and B only.