For anyone unaware, Public Events spawn in the overworld area of Destiny 2 across all its planets. Guardians who participate in the events are invited via the HUD to join in, and upon completion of said event, are rewarded. Interestingly, many events have a Heroic version which is trigger by a player undertaking a specific action. Heroic events are much more difficult, but offer far greater rewards. Here's the details on how to trigger them all. 

Cabal Excavation 

Location: Earth (EDZ)

During this Public Event players must capture a Cabal drill. After around 60 seconds or between 30 and 35% capture progress, a Thresher will appear. Simply destroy the aircraft to trigger Heroic mode: if you don't kill it the first time, it will eventually loop back overheard a minute or two later. Once done, the event's objective will change, requiring you to kill a Cabal Centurion. Unsurprisingly, the most challenging aspect of triggering this Heroic Public Event is bringing the Thresher down. Ideally, you want to use heavy weapons and sniper rifles in order to deal massive impact damage.

Destroy The Arsenal Walker 

Location: Titan

When the Arsenal Walker spawns you'll need to destroy its legs in order to weaken it. Once it has stumbled from damage it will eject several shield cores. You'll need to collect these from the ground and place them into the generators that are shielding the Scorch Cannons. In total you need to do this 6 times, with the Arsenal Walker dropping three cores every time it's weakened. Once you've disabled all three shields, a Devil Walker will appear: kill it to complete the event. 

Disrupt Vex Construction 

Location: Nessus

Part of the Spire Integration Public Event, Disrupt Vex Construction involves preventing the Vex from sacrificing themselves. To trigger the Heroic component, during the Disrupt Vex Construction phase of the event, you'll need to stand within the boundaries of three plates for a period of time. When inside, a timer will begin to count and as long as those you're doing the event with remain within its circle, you'll trigger the Heroic version. 

Ether Resupply

Location: Earth (EDZ)

A simple event that requires you to bring down a large Servitor. Accompanied by small Servitors, which shield their overlord, your job is to kill these smaller varients. Once you have, more will spawn (3 in the second wave). You'll need to kill these quickly in order to trigger the Heroic mode. Be aware that if you don't kill all the second wave within 30 seconds (or there abouts) you'll fail the opportunity to obtain the Heroic event.

Glimmer Extraction

Location: Earth (EDZ)

During this public event you need to stop The Fallen drill teams from mining Glimmer. To trigger the Heroic version, simply ignore the drill teams at each location and instead, destroy each machine near the drill. Once killed at each location, you then need to protect the piles of Glimmer to complete the Heroic event. 

Injection Rig

Location: Earth (EDZ)

The Injection Rig public event sees the Cabal drop a rig into an area. Before you kill Infiltrator Valus, you need to destroy the open heat vents that open on the craft. Opening in phases, there's three at the top, middle and one at the bottom. You need to destroy them all to trigger the Heroic event.

Taken Blight

Location: Earth (EDZ) 

A fairly easy event that can be triggered quickly. During the event three Blights will spawn, with each projecting a bubble around itself. You need to pick up a buff (“Blight Receding") by stepping into the bubbles, before damaging the large Blight orbs overhead. You can apply the buff to yourself simply by stepping back inside a bubble, allowing you to dip in and out to bring down the Heroic trigger quickly. 

Witches' Ritual

Location: Titan

Only occuring on Titan, Witches' Ritual requires you to kill two Hive Wizards. Simply stand in the circular green pads and destroy the crystals above where the Witches spawn (top left and right corners). Once broken, defeat the new enemy who appears.