I agree with the idea of common sense. Without it people are going to get hurt on a scene. Honestly all tools on an apparatus have thier particular importance. From what I have seen, on our department, the importance lies in training and the apparatus that we need. Being from a very rural area, the truck that we use the most is our tanker, and the piece of equipment used most is the dump tank for water supply. This is crucial in our area and will be one of the determining factors on whether you lose property or not.
It depends on whether i'm on the aerial or the engine. If I'm on the engine I tend to grab the halligan and an axe, but if I'm on the aerial I'm more apt to grab the LA trash hook(now that we have them) or the irons.
As a Captain I take a closet pike on residential fires or a pickhead axe. You can pull ceiling with both and it helps with forcible entry. Most Capts on our job don't carry a tool, but I do. I stash it in my belt and it gets used often by me or a firefighter who forgot to bringa tool.