Is there any U.S. / FAA regulation that could apply for aviation software?
For simulators you have Part 60 of CFR 14

For airborne software the FAA has published AC 20-115, but the main document that refers to is the FAA/EASA RTCA DO-178/ED-12 currently at the "C" version: Software Considerations in Airborne Systems and Equipment Certification.
If you want to certify (and thus commercialize) your software for flight, this is the main document to read.

Depending on the scope of the software and the design/development process there might be different supplements (see here):


  
  DO-330 Software Tool Qualification Considerations
  DO-331 Model-Based Development and Verification Supplement to DO-178C and DO-278
  DO-332 Object-Oriented Technology and Related Techniques Supplement to DO-178C and DO-278A
  DO-333 Formal Methods Supplement to DO-178C and DO-278A
  


It has been prepared by the RTCA/EUROCAE joint committee and approved by both FAA and EASA under different names, but the content is the same.

Note that these documents are not freely available.