To find out what stairs you can currently challenge someone in, first go to the Main Stairs Page, then click on "Show your stairs only" link. The ones in BOLD are ones you can make a challenge in. .
Why there are ladders (for example as in http://brainking.com/en/Stairs?trg=3&tri=223, and there are others) where there are two players, no games playing, no games recently played against each other and yet you have no player to challenge?