Do you choose targets for a spell when it goes on the stack/chain or when the card resolves?

Ruling: All targets for all chain items are chosen when the spell is added to the chain, not when it resolves.

Nuances:

• Some cards like Falling Star and Icathian Rain don't actually target themselves - they create separate triggers once the spell resolves, and those triggers declare their targets when being added to the chain
• Cards that say "Do This:" create a separate trigger once the spell resolves, and that trigger will declare any targeting upon being added to the chain
• For Hidden Blade specifically, the target is chosen when it goes on the chain, but "its controller" is determined at time of resolution - if the targeted unit can't be found at resolution (like if it was returned to hand), nobody draws cards
• The targeted unit must still be "a unit at a battlefield" at resolution time for Hidden Blade's draw effect to work
• Replacement effects like Zhonya's or Sett can allow you to keep your unit and still draw from Hidden Blade