Below, we'll be discussing methods of play for Subtlety in 7.1.5.
Authors: @Sathrovarr and @Elving.

Disclaimer:

The goal of this guide is not to be comprehensive. There are other places where you can find answers to questions like this: What are my stat weights? What enchants should I use? What path should I take for my artifact tree? That information is redundant at this point.
For reference to the aforementioned, check out Riff's guide or Celinamuna's guide.
Our goal is to provide some information for those looking to learn, or improve, at playing subtlety in 7.1.5 with tier, Anticipation and dps-specific legendaries - Mantle and Boots, specifically.
Sub is not a very popular spec, so unfortunately there is a relative dearth of information available. The idea is to provide some tidbits that aren't easily located elsewhere.

Onward.

Rotation:

The rotation for subtlety is simple in a broad sense, but gets more complicated rather quickly, depending on your legendaries and the situation. Instead of going into excruciating detail on each ability and rotational aspect (again, use Riff and Celinamuna's guides for that), I'd rather provide some nuggets that are helpful to maximizing your damage.

  • Shadow Dance is the key to your rotation. Never, ever sit at 3 charges. Use charges early and often; the rotation feeds into itself. Dance provides more combo points via Shadowstrike, which leads to more finishers, and finishers reduce the cooldown on Shadow Dance via Deepening Shadows. It's almost similar to the old Combat rotation in this way.

  • Keep in mind Shadow Dance CANNOT be used off the global cooldown. This is a deeply irritating aspect of new Shadow Dance. What it means is that you will try and queue your abilities ahead of time by pressing one of your dance macros, but if you press it too early and don't push the button once your GCD is ready, Shadow Dance will not go off. This happens to even experienced sub players, but there are ways to mitigate it. I find the best way to avoid this is to just really mash your macro button to be certain dance goes off. So long as you don't mash it so hard it bleeds into the next GCD, you won't suffer any consequences from this. There's nothing worse than realizing during an important high damage period that you've cast Backstab instead of Shadowstrike because Dance never went off.

  • Do not spam Backstab. Do not press it the minute it lights up and you have the energy available. Backstab is purely useful as a way to prevent energy capping. It is dramatically less efficient than SS, and we want to limit the number of Backstabs we cast. If you're having trouble, check your logs and see what your ratio of SS to Backstab is. This ration will change a lot based on gear and the exact fight, but you should be shooting for 2:1, at the very least. Once you acquire more gear and get more comfortable with the rotation, 3:1 and beyond is very much achievable.

2pc Bonus & the Pandemic Threshold:

  • With the 2pc bonus, Nighblade's pandemic threshold changes from <5 secs to <7.8 secs. Therefore, it's safe to refresh NB at <7 secs.

4 pc Bonus & Anticipation:

  • Once you acquire your 4pc bonus, Anticipation is the way. The bonus leads to ridiculous combo point overflow, and it's much easier to handle with a buffer of 10 combo points. Unless you're aiming to abuse the Finality bug (feature?) - we'll discuss the Finality bug later. Don't try to finish at exactly 10, or you'll end up wasting combo points. Finishing at around 7+ seems to be a decent compromise point.

Shadow Technique Tracking:

It seems like a pain, but it's important. Shadow Techniques may seem random, but it isn't. The rule is it cannot proc before 3 auto attacks have fired, at 4 auto attacks it has a 50% chance to go off, and if you reach 5 auto attacks without a proc, it has a 100% chance of going off. Good friend of the sub community, Deimos, was kind enough to create a weak aura to track this. We can use the semi predictability of this to our advantage, explained more in the next section.

  • One big thing that sub rogues deal with is what I call the "4 combo point problem". The situation is this: You Dance at 0 combo points, SS twice, and you end up with 4 combo points. This is a conundrum. Typically, we'd want to finish here, because the usual dance cycle is SS twice, eviscerate, then SS twice more. You can theoretically SS again to get to 6 combo points; there's nothing wrong in the abstract with SS x 3, Evisc, SS. But if you SS to 6 combo points and get a Shadow Techniques proc, those 1-2 combo points are entirely wasted. If you try and finish at 4, and don't get a proc, then you will potentially miss out on a Relentless Strikes proc; it's a 20% chance per combo point, so if you're finishing at 4 there's a 20% chance you don't get energy back on your finisher. This is absolutely a killer. Depending on your timing and haste, if you fail to get a Relentless proc, you might not be able to SS again for the entire Dance, wasting 2 whole GCDs that should be filled by SS. That's a disaster.

So the solution, while imperfect, is to track your Shadow Technique procs carefully. The provided weak aura is by far the best way to handle this. If you're sitting at 4 and you will get a proc within the next GCD, then you know you can eviscerate. On the other hand, if your next proc is a ways off, then go ahead and SS, eviscerate, then SS once more to finish off your dance. This problem is much easier to solve and to avoid with Anticipation. Keep in mind that the SSx2 > eviscerate > SSx2 cycle is not sacred. With Anticipation, you have much more flexibility. So long as you fit in 4 SS's and don't waste combo points, the order doesn't really matter.

Finality:

Understanding Finality is another important aspect of playing Subtlety. For Nightblade, this is crucial. Finality applies a buff when either Nightblade or Eviscerate it used that increases the damage of the next Nightblade or Eviscerate. We can utilize this in conjunction with the Pandemic mechanic to ensure higher up time for our buffed Nightblades. Essentially, when you have the Finality Nightblade buff up, clip your Nightblade right at the Pandemic threshold (30% of the total duration of the debuff, and this will change depending on tier/talents/etc.). When you don't have a Finality Nightblade buff up, refresh it right at 0.

  • There is also a... 'feature' (possibly a bug, it is unclear) regarding Finality + Anticipation + Eviscerate. Right now, if you don't have the Finality buff, and you cast Eviscerate with 10 combo points via Anticipation, the Finality buff applied to your next Eviscerate will be 40%, instead of the usual cap of 24% with Deeper Strategem (Finality buff is 4% increased damage per combo point spent). This has the potential for some abuse. You can prioritize having Finality Eviscerates with a 40% buff ready to go during high leverage windows of damage; this is particularly useful if you have the legendary shoulders.

Because we care:

  • Be mindful of your Master of Subtlety up time. It's something a lot of people (myself included) kind of lost track of for a bit, because most people were running Weaponmaster. Now (partially due to the ridiculousness of Mantle of the Master Assassin, and also the up time we can achieve on the buff) Master of Subtlety is back in vogue. It's certainly not the most important part of your rotation, but maximizing the buff is still better than not. Anticipation makes this easier. If you just finished a Dance cycle and you're sitting at 1-2 combo points, a lot of times you'll be ready to Dance again right away. During high damage periods, like during your potions or other cooldowns, chaining them together makes plenty of sense. But during lulls in the middle of the fight, there's nothing wrong with throwing in 1-2 backstabs before going back into Dance. This will space out your dances a bit more and stave off the inevitable point where you run out of charges. Spacing out the Dances a bit more, will, in turn, increase your Master of Subtlety. It's a small thing (a.....subtle thing) but it's worth being mindful of.

  • Be careful replacing Energetic Stabbing relics. These relics (thanks, Blizzard!) don't drop at all in Nighthold, but they modify our best trait, which is a chance for SS to refund energy on use. The fact that they don't drop in NH means that NH relics with lesser traits are going to start outpacing them; the Demon's Kiss (Nightblade damage) relics from Tichondrius for our Fel relics are likely culprits, as well as the Gutripper relic (Eviscerate crit chance) for our Shadow relic that drops from Gul'Dan. This is more feelycraft than real advice; following simcraft and hard numbers is almost always the best advice. But Energetic Stabbing relics are enormously helpful in smoothing out our rotation, and giving you an energy buffer to ensure you can fight 4 SS's into each Dance. If sims tell you a different relic is much better, then you should use it. But if a relic is roughly comparable to an ES relic, or even a marginal upgrade, I would consider sticking with ES.

  • Helpful macros? Refer to Riff's guide.

Openers:

Below, we'll offer some openers for you. All of these openers assume 4pc bonuses and will be updated in the coming weeks as more testing is done to further our understanding of where DoS should be used, optimally:

4pc + Mantle:

ShBlades > SS > SS (> SS a 3rd time, if you can) > NB > Dance > SS > SS > Evis > SS > SS > Vanish > Evis > Evis > SS > SS

  • Sprint after first Vanish or where appropriate during encounter and then on CD.
  • Always enter into Vanish/Sprint with >5 cp's so that you can immediately Evis.

4pc + Mantle + DoS:

ShBlades > SS > SS (> SS a 3rd time, if you can) > NB > Dance > SS > SS > Evis > SS > SS > Vanish > Evis > Evis > DoS

  • This allows for 1 more Mantle GCD at the end where you can start another Dance.
  • Sprint with DoS, then on CD w/ DoS throughout the fight afterward - macro DoS to Sprint.
  • Always enter into Vanish with >5 cp's so that you can immediately Evis.

4pc, No Mantle:

ShBlades > SS > SS (> SS, if you can) > NB > Dance > SS > SS > Evis > SS > SS > Dance

  • Vanish after 3rd Dance or when you reach 1 charge of Dance and then on CD.
  • Sprint after first Vanish (or where appropriate) and then on CD.

Sprint Usage:

Once you've attained 4pc and you're utilizing Anticipation, we stress the usage of Sprint as a dps cooldown. It's been asked many times where one should Sprint and how.

  • Sprint during transition periods of an encounter, i.e., Odyn entering the fight in ToV.
  • If it's necessary to do so, use Cloak before entering into Sprint/Vanish.
  • Use the provided Sprint macro.

#showtooltip /stopattack /cast Sprint

With gratitude, contributors to this reference, as well as their own guides, and continued contributions to the Rogue community are:
@Gray_Hound, @Sathrovarr, @Jaxu, @Elving, @Riff, @Deimos and @Aethys.