Is there a reg that covers whether or not a single soldier can rent an apartment?

This is very much a station to station situation. You can start by asking your local housing office and command.

Make sure you're not in one of those "must live in barracks" situations. If not, you can opt to live out in town. Also run the numbers like what BAH do you get, or not, etc. Then run the numbers for your 3 squares in the galley. See what leaves you with more money in the wallet. That goes into the overall decision as to what you choose. When I was single ENL, I didn't have an option other than live aboard ship or in port barracks. I wouldn't get any money for housing. Think that changed a long time ago but better safe than sorry.

Also if you're going to share a unit out in town with some other MILs, just remember if they go into barracks slob mode, you have far less recourse than on base. Think it through and live on....

As asked? No, there is not that i know of in the US.. NON CONUS commands may have local regs that bar such a action.

If you're asking if the Service should have to pay you to live off post then the answer is yes, there are regs that cover that.
Could you be a bit more specific in your question? Provide a bit of context?

In general you may spend your personal funds on anything that is legal and does not constitute breach of good order and discipline.