We don’t often have a Troublesome triplet, but these three do go together and I’ve been asked about them a few times now. This is another one where the American English is a bit different (enquire is more common in British English, inquire in American English, which matches use of ensure and insure), so we’re sticking to British English here. The Dictionary doesn’t distinguish between them that clearly, but my “special” dictionary for Editors and Writers does, so that’s what I’m sticking with!
Enquiry – a request for information. To enquire – to ask for information. “I would like to enquire as to the price of this article”; “Please enquire within for information on our rates and services”; “Directory Enquiries”
Inquiry – a formal investigation (e.g. by the police, the courts, etc.). To inquire – to make a formal investigation. “An official inquiry has been launched into the murder of the policeman”; “The Select Committee are inquiring into the expenses scandal.”
Query – a question. To query – means to question, to ask a question about. “She queried the amount she was asked to pay”; “I have a query about the expenses you’ve claimed.”
“I have an enquiry about the outcome of the police inquiry; can you answer my query about paragraph 4?”
You can find more troublesome pairs here.