1) Signing bonuses are signing bonuses. Those, you should pursue anyway because they are low hanging fruit.
2) To decide what card to charge to after completing all signing bonuses (in the absence of bonus categories), it depends on how you value your miles. You could be using a generic travel card like a Barclay Card arrival to earn 2.2% back in the form of travel credits. Or you could be earning 1 Chase UR (or 1 US Airways Dividend Mile) per dollar.
What's preferable? Depends on your valuation.
Here is one blogger's valuation, along with justification on why (complete with sample searches, etc). Your valuation may be higher since you want to use your miles towards business international seats, rather than domestic economy seats. I've found that to obtain a proper valuation for each type of reward currency, you have to do a lot of award searches. There's no substitute for experience, and in this hobby, I think that doing award searches yourself is the input that creates that experience.