Normally, as a rogue, I'm allowed to apply Sneak Attack Damage to any one of my attacks made on my turn. Normally, it doesn't matter which attack I apply it to; often only one attack connects anyways, and it's the same Xd6 damage regardless of the attack...

... Unless one of them is a critical hit.

A successful critical hit means the damage on my sneak attack dice will be doubled, which means I'm incentivized to prefer applying the damage only on a successful crit, and the wording on Sneak Attack doesn't specifically say I'm required to apply it only to my first landed hit; only that I may apply it to one attack that has hit.

But there's one issue: I roll my attacks one at a time. So consider the following play-by-play, using Two-Weapon fighting with Short Swords:

1. I roll to attack my target (Action). AC 16, I roll 13 + 7 = 20, a successful hit. I apply my sneak attack damage.


2. I roll to attack my target (TWF Bonus Action). AC 16. I roll 20 + 7 = 27, a critical hit. But I've already applied my Sneak Attack damage, so my sneak attack damage can't benefit from this crit.

In this scenario, I've lost the potential extra damage from critting with Sneak Attack. Now consider the same scenario, but with a different strategy, and different die rolls:

1. I roll to attack my target (Action). AC 16, I roll 13 + 7 = 20, a successful hit. I could apply my Sneak Attack, but I want to save it for a critical hit, so I don't apply it.


2. I roll to attack my target (TWF Bonus Action). AC 16. I roll 8 + 7 = 15, a missed attack. I can't apply my Sneak Attack damage to this attack.

So the question is, now that I know my second attack isn't going to connect, am I allowed to retroactively apply my Sneak Attack damage to my first attack, or do I simply lose the damage from my Sneak Attack as a result of not applying it to my first attack? Am I able to optimize my sneak attack damage by being picky about which attacks I apply it to, or am I incentivized to simply apply it to my first successful hit to avoid risking losing the damage altogether?

You cannot defer the decision.

You must decide whether or not to apply Sneak Attack immediately after making a successful Attack Roll.

Official Ruling

Jeremy Crawford, voice of Sage Advice and the official voice of rulings by Wizards of the Coast has this to say:

You decide whether to use Sneak Attack when a qualifying attack hits.

source

On further questioning...

Q: sneak attack+ multiple attacks, if 1st hit but hold sneak to see if 2nd crits & 2nd misses, can they retro apply it to 1st?

A: Sneak Attack doesn't give you the ability to turn back time.

source

So, the procedure is this: Roll a d20. Does the attack hit? Yes? Now you must choose whether or not to apply Sneak Attack damage.

You cannot roll an attack, then roll damage, then roll an attack, then roll damage, then decide to apply Sneak Attack to one of the prior attacks.

Logic

Same as above.

Does it make any sense for your character to stab someone...then stab them a second time...and go "Oh, wait, I did a better job the first time, I'll stab them harder on that first stab"?

No. Because you already finished stabbing them.