Well, what is interesting and how to decide what is interesting depends entirely on your particular application... usually it will have to do with which lines are active (i.e., have changing logic levels, or voltages exceeding some threshold, or have certain patterns or logic conditions characterizing them).
As an example from my research, you might have a set of detectors which only produce signals some of the time, when a particle goes through them. If the voltages of the signals on some set of detectors exceed some threshold within some time window (the condition might be simple or complex), then a ``trigger'' is generated. The trigger signal can be used to tell a multiplexer to send data through for the channels that are active.
Or, say you want to send a phone signal through to a particular phone number. Then you would address the lines corresponding to that number in order to pass on the information. (Of course in an actual phone the circuits are more complicated that our humble little MUX box. But the general concept applies.)