Let’s say you have Feather, the Redeemed in play and cast Double Cleave targeting Feather.
Your opponent casts Misdirection changing the target to his or her creature instead.
Misdirection resolves.
Double Cleave resolves.
Let’s say you have Feather, the Redeemed in play and cast Double Cleave targeting Feather.
Your opponent casts Misdirection changing the target to his or her creature instead.
Misdirection resolves.
Double Cleave resolves.
What happens to the Double Cleave?
It gets exiled and can be expected to return to its owner's hand at the beginning of the next end step.
The ability of Feather the Redeemed triggered when its controller finished casting Double Cleave. It doesn't "untrigger" when the target changes during Misdirection's resolution.
603.3. Once an ability has triggered, its controller puts it on the stack as an object that’s not a card the next time a player would receive priority. See rule 117, “Timing and Priority.” The ability becomes the topmost object on the stack. It has the text of the ability that created it, and no other characteristics. It remains on the stack until it’s countered, it resolves, a rule causes it to be removed from the stack, or an effect moves it elsewhere.
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Why bother with mere rulings when so many answers can be found in the Rules?
Feather's triggered ability cares what the instant or sorcery spell targets only at the time the spell is cast (C.R. 601.2i, 603.2). The ability doesn't care what the spell targets at any other time, including when that spell or that ability would resolve or when that spell's targets change (Feather's triggered ability doesn't have an intervening "if" clause, for example [C.R. 603.4; compare that ability with Ink-Treader Nephilim's]). (In general, however, a spell with targets will fail to resolve if all of them are illegal [C.R. 608.2b].) See also this thread, this thread, this thread, and this thread.
As a result, if Double Cleave targeted a creature you controlled as you cast that spell, Feather's triggered ability will trigger for that spell, and the ability will still get to resolve even if Double Cleave's target is no longer a creature you control, or becomes illegal, in the meantime (see also C.R. 113.7a).
On the other hand, if Double Cleave didn't target a creature you controlled as you cast that spell, Feather's triggered ability won't trigger for that spell, even if Double Cleave's target then changes to a creature you control.
EDIT (Dec. 29): Edited.
EDIT (Sep. 10, 2020): Correctness edit in view of recent rule changes.
Your opponent casts Misdirection changing the target to his or her creature instead.
Misdirection resolves.
Double Cleave resolves.
What happens to the Double Cleave?
It gets exiled and can be expected to return to its owner's hand at the beginning of the next end step.
The ability of Feather the Redeemed triggered when its controller finished casting Double Cleave. It doesn't "untrigger" when the target changes during Misdirection's resolution.
As a result, if Double Cleave targeted a creature you controlled as you cast that spell, Feather's triggered ability will trigger for that spell, and the ability will still get to resolve even if Double Cleave's target is no longer a creature you control, or becomes illegal, in the meantime (see also C.R. 113.7a).
On the other hand, if Double Cleave didn't target a creature you controlled as you cast that spell, Feather's triggered ability won't trigger for that spell, even if Double Cleave's target then changes to a creature you control.
EDIT (Dec. 29): Edited.
EDIT (Sep. 10, 2020): Correctness edit in view of recent rule changes.