Shadow Flygon

Shadow Flygon is an offensive improvement compared to its non-Shadow variant, but as more Pokemon gain Shadow variants Shadow Flygon will be left in the dust again. On defense its double weakness to Ice-type attacks does it no favors. In short: it's a strict down-grade to Garchomp.

Best PvE Offensive Moveset

• Dragon Tail should be paired with Dragon Claw. This set has better neutral coverage and damage than the Ground-type set.
• Sand Attack should be paired with Scorching Sands. 
• Mud Shot is okay, but it's a bit behind.
• Stone Edge may have a few niche uses, but is generally the worst option.
• Boomburst, Earthquake, and Earthpower* are inferior.

Best PvE Defensive Moveset

• Dragon Tail is Shadow Flygon's only viable fast move on defense, with triple Mud Shot’s power.
• Dragon Claw is Shadow Flygon’s best charge move option, having 3 charge bars and an early damage window.
• Earth Power* is a decent second choice, as is Scorching Sands
• Stone Edge has counter coverage, but Shadow Flygon’s bulk isn’t well suited for 1-bar moves.
• Sand Attack, Earthquake and Boomburst are underwhelming on defense.

Dragon Tail is a solid choice after the buffs from the GBL Season 8 update, with strong STAB damage output and average energy gains. Mud Shot provides very fast energy gains to help Shadow Flygon spam its powerful charge moves, but has very poor damage output. Sand Attack trades off a little bit of energy generation for a little more raw power, making it viable as a choice as well.

Dragon Claw sees a lot of use on Shadow Flygon, which is often used to apply heavy shield pressure and bait shields. Though its base power of 50 and 1.4 DPE (Damage Per Energy) isn't anything spectacular, the frequency of which it can be used is noticeable. Dragon-type opponents also dislike being abused by constant Dragon Claw spam. 

Scorching Sands has a very respectable DPE (Damage per Energy), a relatively low energy cost, and a solid chance to drop the target's Attack on use, making it an optimal choice. Earthquake has the potential to deal huge damage to the opponent if it's unshielded (and if the opponent doesn't fly), making it also worth considering. Stone Edge adds coverage to Flygon and helps deal with its double weakness to Ice-type Pokemon, in addition to common Flying-types. 

Earth Power, Shadow Flygon's Community Day special move, has a DPE very similar to Scorching Sands but with higher energy cost and low odds of cutting the target's Defense, making it the lesser option in most cases.