Let me start with a disclaimer: Any weapon is useful. A piece of hard material with a sharp edge can be dangerous. Plenty of people have been injured or even killed with nothing more than kitchen knives, broken bottles or a pointed stick. Any statements I make below pertain to optimal tool design for a task. The added advantages of a certain blade geometry may not add up to much in the grand scheme of things.
That having been said, I believe that the optimal knife for fighting/self-defence is double edged. For fighting, knives have to be able to stab, slice and (if large enough) chop. Long standing common knowledge is that the point on a double edged blade penetrates much easier than a single edge because it is thinner. Double edges also allow you to cut with both the inside and outside of the knife without indexing the blade, which I have found handy especially in reverse grip. The sharp second edge also makes it more difficult to disarm you by manipulating the knife itself (i.e. disarm techniques that put pressure on the unsharpened edge).
The double edged blade is not as good for utility. The thinner tip is more fragile. As has been said earlier, you cannot put finger pressure on the back of the blade if it is double edged (well maybe you could, but it would be messy

). Also, you cannot use as shallow an angle for the bevels if they have to meet at the center of the blade, so a sturdy double edged blade is not as good for deep cuts.
If I personally had to choose one knife to do both self defence and heavy duty utility, I would pick a single edged drop point or clip point blade design. I base this decision on the idea that the blade would see more duty as a utility knife than as a defensive knife (hopefully). The drop or clip would still give enough of a point for stabbing if needed but would not be as fragile as a double edged point for it's main intended function. A penetrator tip like the one used on Busse/Swamprat knives might also be a good compromise.
The most important aspect of weapons self defense is mentality. It don't matter what kind of self defense tool you use if you do not have the resolve to use it.