PHB says 'once per turn,' so, it would seem that a Rogue with Sentinel, for example, could "consistently" deal Sneak attack twice per round (whether the enemy triggers an OA or the melee weapon attack provided by the Feat).
So, can rogues deal sneak attack damage with any weapon attack outside their turn? (obv. one that meets the standard requirements--finesse, adv or adj ally)
Yes!
The verbiage 1/turn definitely allows for the use of Sneak Attack out of turn for the rogue if they somehow get a reaction attack (either granted by an ally or by an opportunity attack).
Note that this basically limits them to 2x per round since you only get one reaction. But yes absolutely. Note that all of the normal restrictions on SA apply, they have to have advantage or an adjacent ally, and they have to be attacking with a finesse or ranged weapon.
Sneak Attack. ...Once per turn, you can deal an extra 1d6 damage to one creature you hit with an attack... [some restrictions may apply]. (PHB p.96, "Sneak Attack")
There are two important phrases in that description I've highlighted.
Once per turn. The writers could have said "on your turn." They didn't, so it doesn't have to be your turn*. Compare, for example, with Reckless Attack (PHB p.48) which says "when you make your first attack on your turn.... (emphasis mine)" A Barbarian clearly cannot both Recklessly and Opportunely attack.
You hit with an attack. The writers could have said "when you take the Attack action." They didn't, so the attack you make as a reaction (opportunity attack) is viable. Compare, for example, with the Battle Master's various Maneuvers (PHB p.74): some trigger "when you make a melee weapon attack on your turn," some "when you hit a creature with a weapon attack," some "when you take the Attack action." These subtle differences in applicability are ones that the designers are well aware of, and use to balance features.
(Jeremy Crawford also says so, albeit tersely, in this Sage Advice Exchange: http://www.sageadvice.eu/2015/06/18/sneak-once-per-turn/. But he didn't need to, since the rules already said as much.)
* - In case there's any confusion, look to PHB p. 189, The Order of Combat: "during a round, each participant in a battle takes a turn." So other people's turns are a turn, meeting the restriction in Sneak Attack.
You only get one turn per round. A turn is comprised of an action, movement, reaction, free action and bonus action. Sneak attack let's you add bonus damage ONCE per turn. If you decide to use it with your action, bonus action or reaction is up to you. If you decide to use it during your official turn or during someone else's turn (reaction/ready action) it's still used.
More info:
A sneak attack:
Once per turn, you can deal an extra 1d6 damage to one creature you hit with an attack if you have advantage on the attack roll.
A turn:
A round represents about 6 seconds in the game world. During a round, each participant in a battle takes a turn.
Reaction:
When you take a reaction, you can’t take another one until the start of your next turn. If the reaction interrupts another creature’s turn, that creature can continue its turn right after the reaction.
