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. .
MadMonkey: I suggested a year ago or so, having a red number in parentheses next to the Stairs link, that would be clickable. This number would be the number of stairs you are in where you can make a challenge. When clicking on it, it would take you to your personal list of stairs. (The same page you get when clicking "Show me my Stairs Only"), where you would see those you can make a challenge in, in bold. Obviously, there will be occasions (relatively rare, I imagine) that you click on this number, and there may no longer be a challenge available. I guess that depends on how often that red number can be updated. I imagine pretty quickly, since the other red numbers are updated instantly.