Hi,
When I play FF, I just carry what ever is in my pocket. Which is generally a slip-joint of some kind, (and generally plain high carbon steel to boot). As a FF, your team should never get off the truck without at least a set of irons. When you have close to a half-million dollars worth of truck and gear on scene, you should have what you need to do the job at hand. And if you don't, you should be calling for the people who do.
When I work as an EMT on the ampalamps, the last thing I ever want to see is a sharp pointy tool next to me and my patient. At least one of us is already having a bad day and I don't want to deal with the possible extra stress that could be caused by the sight of the knife. Or the risk that some adrenalin-pumped assistant might cause an injury. If there is cutting that needs to be done, my EMT shears will do it.
That said, most of the FF and EMT's I know carry either a multi-tool, or a small folder that fits easily and unobtrusively in a pocket. They are handy around the station and in the garage. Us older guys tend to favor slippies and the younger guys smaller folders, either frame or liner locks.
In 15+ years of doing this, I have never needed a knife on scene. Though I will say that I've used my multi-tool a couple of times. I had to pop the hinge pins out of a bathroom door once. And I used it to unship a toilet to make it easier to remove a patient from a bathroom. And then I promptly left it behind, never to be seen again. But, I could have just as easily used the tools from the toolbox on the rig to do the job. I was just too lazy to get them.
Remember, Scene size up and Scene Safety! I don't want to read about anybody on the secret list.
dalee