Shadow Beedrill

Beedrill lives up to its initial use in the original games flawlessly: an easily obtainable pokemon that's easy to raise and help bolster an early team. Sadly, its best role is nothing but a stepping stone to greater things in the game as any real collecting will yield better poison and bug types and leave Beedrill outclassed. On defense it's got a tiny bit of merit early on due to double-resisting Machamp's signature attacking set, but by that same token its entire moveset is hard-walled by Steel types.

Best PvE Offensive Moveset

• Bug Bite* and Infestation have roughly similar performance, with Infestation having better damage and Bug Bite better energy gain, though the legacy Bug Bite pulls ahead just slightly. 
• Poison Jab with Sludge Bomb is actually Shadow Beedrill's strongest moveset by a fairly wide margin. Poison Sting falls behind.
• X-Scissor is fast and furious, giving Shadow Beedrill a bug type set.
• Aerial Ace is fairly slow, but if you've just started and are desperate for fighting-type counters, you can leverage Shadow Beedrill's double fighting-resistance and use this. However, even many neutral matchups can out-damage Shadow Beedrill on this metric.
• Fell Stinger looks like a clone of X-Scissor, but it's substantially slower which makes it weaker in practice.
• Beedrill is a bad Bug-type attacker. It's an even worse Ground-type. Leave Drill Run alone.

Best PvE Defensive Moveset

• Poison Sting is the best Fast Move option available for defense, followed by Infestation.
• Poison Jab and Bug Bite* are the above moves, but worse when it comes to defense.
• Drill Run generally stands as the best Charged Move option due to its outstanding coverage.
• Arial Ace is solid overall.
• X-Scissor and Sludge Bomb should only be paired with Fast Moves of the opposite type, though their coverage is pretty similar so it's not a HUGE deal if you want to use them.

When looking at stats, Poison Jab is superior to legacy Bug Bite, though they're different typings and cover slightly different roles. Poison Jab has recently  become a stronger choice than it once was thanks to the presence of Charm users such as Wigglytuff and Clefable. Infestation has the highest gains by a very small margin, but lacks raw damage output. 

X-Scissor and Sludge Bomb gain STAB with Beedrill and solidify its role as a Bug/Poison type. The former costs very little energy to use and can be spammed to bait shields, while the latter packs a harder punch with a higher 1.6 DPE rating. Aerial Ace is an inferior move that doesn't get STAB even if it would offer coverage against Fighting-type Pokemon. The addition of Fell Stinger following the Dec 11, 2019 rebalance is interesting for Shadow Beedrill, but isn't very impactful if it doesn't bait a shield due to its pitiful damage output. Beedrill's Community Day move, Drill Run, gives it the much needed coverage against Steel-types that it's STAB moves lack. It's also quite a powerful move, with 1.78 DPE.