Before you can answer that question, you kinda have to figure out what your asking about. Are you talking about beating the final boss? or just for getting there in the first place? Are you talking about which is the most powerful, or which has the best strategy? I'm gonna try and answer all of these questions, and more. (this is entirely my opinion. If you feel differently, please post your thoughts below)
Types of Weapons:
Common:
-Laser
-Ion
-Missile
-Beam
Uncommon:
-Bomb
-Flak/burst
-Crystal
Which weapon is the best for beating the Rebel Flagship?
This is a very hard question to answer. There are so many strategies for beating the boss, that it would be almost impossible to answer. But I'm gonna try. In my opinion, the breach bombs are the best. They easily kill the Flagship's crew, and can stop the AI repair. They also damage the systems, so overall its a pretty good weapon no matter what. Using this to disable the weapons and shields pretty much leaves the flagship vulnerable to almost anything.
Which weapon is the most powerful?
This is an easy question to answer. Its the Glaive beam. This weapon does so much damage and has such a long trail for how much damage it does. The game even says that its the most powerful weapon! Of course, if you go into the source code and start cheating, you could argue that the boss weapons are the best. Especially the boss missile! The boss missile shoots 3 missiles for the price of none! While this is a valid argument, I'm gonna stick with the Glaive beam. For god's sake it can never miss! You could also say vulcan, and I would agree that its very powerful, but its just not the same as the Glaive beam
Which two weapons work the best together?
Well there's an easy way to answer this. Ion plus laser = good bye shields! But I'm not gonna go the easy route. I feel the best duo is fire and lockdown bombs. You lock all enemies in a room, then burn 'em to death! Yee-haw!

Which weapon is best used for eliminating enemy crew?
A lot of people would go bio-beam. I mean it was literally made to kill the crew! I disagree. While its very powerful, it cant pierce shields. That's a major down side. I'm gonna have to go with my response for the flagship question. Breach bombs for everyone

Which weapon is best used to destroy AI ships?
Breach bomb.
Overall, what is your favorite weapon and in your opinion the best weapon.
Well, that's a hard one... BREACH BOMB