You could mess around with the "timeout" trigger.
Set it to minimum, turn the gain of one channel way up and trigger off the noise on that channel. That leaves you three channels to play with.
I suppose I could hack this behavior by setting the time base mode to XY and feeding a sawtooth wave into the X channel and the probe into the Y channel but it seems like the scope ought to be able to do something like this on its own.
If you're going to do that then feed the sawtooth into a channel and trigger off it. That leaves you with three other channels.
I think the trigger channel can be hidden if it annoys you to see ti on screen.