Shadow Gyarados

Shadow Gyarados is powerful without a doubt. It stands very high on the Water Type DPS charts in particular, making it a great choice for most players. In fact, Shadow Gyarados actually out-damages Mega Gyarados when not counting Mega Boosts! If you're lucky enough to find a high-IV Shadow magikarp, then Shadow Gyarados may be worth keeping over purifying for the sake of Mega Gyarados.

Best PvE Offensive Moveset

• Waterfall and Hydro Pump give Shadow Gyarados maximum Water Type damage
• Aqua Tail* is a solid second choice thanks to its lower energy cost and reliability, but the damage drop is noticeable. 
• Bite and Crunch can make Shadow Gyarados acceptable as a Dark Type
• Dragon Breath and Outrage work if you're desperate for Dragon Type DPS
• Dragon Tail* is actually better for this, but it's legacy and not worth an Elite TM to obtain.
• Twister and Dragon Pulse* are not worth using.

Best PvE Defensive Moveset

• Dragon Tail* is arguably the best move for defense, though its Elite status makes it hard to obtain.
• Waterfall is generally just as good.
• Crunch and Aqua Tail* both have a low energy cost and solid neutral coverage, making them great choices for defense.
• Outrage also has great neutral coverage and high power, but it's more expensive and is fairly easy to dodge.
• Twister comes out fast and often, but it's also a weak move.
• Dragon Pulse*, Dragon Breath, and Bite are all inferior choices.

Dragon Breath is a blessing to virtually anything that has it, and Gyarados is definitely no exception. It sports high Damage per Turn and average Energy per Turn, which makes it Gyarados' objectively best quick move despite lacking stab. 

Aqua Tail is an incredibly cheap move that can provide frequent shield pressure. Hydro Pump can threaten KOs when not shielded, but its high energy cost makes it a less consistent option. Crunch provides good overall coverage, decent damage, and a reasonable energy cost, while Outrage is very powerful at a fair cost. This set provides Gyarados with a powerful catch-all playstyle that's hard to come into, and is likely to deal meaningful damage even to Gyarados' counters, while also allowing it to act on opposing dragons while not being hindered by a dragon weakness itself if packing Outrage. Return can be used as a coverage move against Primarina.

You could also go for Bite, with similar performance to Dragon Breath but lower Energy Generation, making Gyarados an aggressive Dark attacker. It is worse against Giratina, however, and with it Gyarados becomes even slower. Waterfall is also not a bad move, and is better against Steel and Fairy Pokemon, but there are generally many superior Water-type attackers to choose from.