When are you allowed to use a call sign that is not the aircraft registration?
For the US, this is covered in AC120-26L, or in a more human-interpretable way from an NBAA article.


  When are you required to use your aircraft registration


By default unless you have a callsign


  Can anyone use whatever call sign they want or is it limited to airlines only?


What you use must be registered. For international flights if you fly frequently enough you can get a 3 letter ICAO designator (SWA, DAL, UAL, etc.)

There are national callsigns which are detailed in that letter, and local callsigns which may apply only to your tower, TRACON, or center depending on who you talk to and what their own rules for issuing a letter of agreement to use it are.