How many times does Sneak Attack damage apply?

Question

In 3.5e, I used to add Sneak Attack damages to all the attacks of my character as long as the target was flat-footed or flanked or...

However, when discussing the feature with wax eagle he mentioned that in 5e Sneak Attack applies once per turn.

I am not sufficiently familiar with 5e terminology yet to understand exactly what that wording entails. So, I imagined the following situation:

• I strike a surprised opponent thrice, twice with my main-hand and once with my off-hand
• then, before my next turn, I also deal one strike to an opponent streaking past (AoO)

In that situation, which of my attacks benefit from Sneak Attack damages ?

• in 3.5e, it would have been all, that does still apply ?
• only one attack per opponent ? (once for my regular turn and once for my opportunity turn)
• only one attack at all ? (opponent of my choice, possibly wasted)
• only the first attack ?

Answer 1

I strike a flat-footed opponent thrice, twice with my main-hand and once with my off-hand then, before my next turn, I also deal one strike to an opponent streaking past (AoO) In that situation, which of my attacks benefit from Sneak Attack damages ?

in 3.5e, it would have been all, that does still apply ?

No, firstly, there is no "flat footed" in 5e, so inorder to get sneak attack damage you either need an ally adjacent to your enemy or advantage on your attack roll. For the sake of this question we will assume you have one of those.

Second, there isn't really an easy viable way for a rogue to hit three times, but let's pretend you multiclass rogue and fighter and have enough levels to get 2 attacks plus an off hand bonus attack.

As you said in your question, sneak attack only happens once per turn, so only 1 of those three hits would get sneak attack damage applied to your damage. Presumably the first one.

only one attack per opponent ? (once for my regular turn and once for my opportunity turn)

So this question here is a bit misleading. You can use sneak attack damage once per turn, not once per round. Your opportunity attack happens when it is not your turn, so you can apply sneak attack damage. But only because your turn from your first hits are over.

only one attack at all ? (opponent of my choice, possibly wasted)

Only one attack while it is still your turn.

only the first attack ?

You can in theory decide to not apply your sneak attack damage to your first attack, but you can't do that if you already know the results of your second and third attack.