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. .
plaintiger: If the players are equally matched, the probability of a split is 50%. So, if you are seeing nothing but splits in a lot of matches, that may indicate a glitch somewhere. However, I can't imagine how a bug like that could work its way into the system, since (among other things) the dice generator must be independent of the match format. So, I suggest that what you are seeing is purely the result of chance. In time, every event that is merely improbable must happen.