It's hard for me to say, but my opinions on the three are as follows:
Kostura Cannon: Pair this with an Orkulus/Phase Gate (Entrenchment ONLY, mind you), and you can send in a fleet to wipe out the already-crippled planet defenses. Alternatively, you can fire two at a pair of gravity wells, giving you a makeshift (if risky) shortcut to use at any time. Great for the late game, but not so much when starting up.
Novalith Cannon: The Novalith has the nasty side effect of lowering a planet's population, not to mention putting a serious setback at repopulating. Sure, it can wipe out an enemy planet in no time at all, but the drawback is that you yourself have to also deal w/the lowered stats. Mainly effective for administering the finishing blow to your enemies (and winning), or if you just feel like a little geocide.
Deliverence Engine: The Deliverance is actually more beneficial than one might think at first. For one, all friendly craft in the targeted well get a boost to their attack, which makes for a great "Panic-Button" weapon. Granted, you still have to wait for it to fire, but once it does, woe to the enemies of the Unity. Additionally, with Eyes of the Converted researched, any planets reached by the weapon's culture spread can be seen, if only briefly. The great thing is you can focus fire on one planet or spread it over several - both have their own advantages. Good for pretty much any situation.
For All Superweapons: I've always tried to set them up in the center of the map, or somewhere secure from attack. On the one hand, I love the Novalith for its sheer Kaboom factor (and WHAT a boom, indeed), but the Deliverance is a bit more flexible in what you can do with it. Even then, the Kostura's ability to create a phase gate wherever you need it is a bonus in and of itself, but you also need one to get back.
So, in closing, I believe my choices are:
Side question: Does the Kostura kill mines as well? I've never been able to see for myself...