paperwork...
They never asked for anything from my prior companies. BUT, if you go with the company I feel is one of the best for AJX, ( ok, standing by for the hammer ), Crew, which is an American company, will run you through the background check, which some company from the US, tries to contact them. When they do not receive a response, then they will ask you for some kind of proof that you worked there. Which is easy enough if you have a copy of your contract (and some pens and t-shirts etc....)
Other than that, beyond a shadow of a doubt, you will be much happier for the long and short term with Air Japan than Jetstar or Peach. That isn't my opinion, it's absolute fact.
Good luck and ok boys, I'm ready for the hammer......