What cell phone service is available in St. Thomas? Can you rent one for use while there, and if so where and at what cost?

AT&T is there. Same as in the continental US.

Also Sprint. But AT&T has better coverage.

What about Verizon?

$$$$ ouch

Spent last winter in and around USVI, check with your carrier because most US cellphones work with your current plans, no extra charges unless you are on St John and roam onto the BVI cell system. You can also buy phone cards all over the island. Some signals are stronger on different parts of the island depending on your location, same as you would have every where. There are also some good posts on tripadvisor, worth reading

Verizon is roaming. As sail2wind said: $$$

There are places on St Thomas and St John where all you can get is a BVI tower. However, there are places in the BVI with line of sight to Crown Mountain and you can get great reception, internet (4G), etc.

Maybe buy an AT&T 'burner' phone?

I have TMobile and in the USVI it's free. In the past if I was in the BVI or North shore of STJ and picked up a BVI tower the charge was $4.49 per minute. Just recently it dropped to $.20 per minute in the BVI. It's a huge savings for me since I'm down there 4-5 times a year split between Tortola and STJ.

Sprint actually even works from Cane Garden Bay if you can see St. Thomas! FWIW, unusable on Culebra except in Dewey.

I have a Tracfone and am very happy with it. 39.95 at Target and the phone card for 230 minutes is 30.00 at CVS. It has great service, voicemail, textmessaging, camera. etc. Everything you can get on a smartphone. I highly recommend it as a second phone for the USVI. I am a Verizon subscriber and do not even use my Verizon acct in the USVI.