The People's Hero - Destiny 2

Complete a Heroic Public Event - 40

Public Event happen all over the each world all the time, you allow the Public Event to upgrade to a Heroic Event you need to complete a minor task, while completing the Public Event. You'll know when the Public Event has upgraded because a message will read "Heroic Event Started". If you complete the Heroic Public Event, you will receive more glimmer and better loot.

EDZ (European Dead Zone):

Fallen Glimmer Mining - Destroy the Small Drilling Machines, rather than killing enemies.

Ether Supplement - Destroy the smaller Servitors, rather than the Elite Servitor.

Injection Rig - Destroy all 3 Glowing Smoking Vents on the Rig.

Titan:

Fallen Walker Attack - Once the Walker has fallen to it's knees, it will drop 3 Arc Orbs, the Arc Orbs can be used to destroyed the shields around a Scorch Cannon - you will need 6 Arc Orbs. Doing this will spawn a 2nd Fallen Walker.

Witch's Ritual - Kill both Witches, then destroy the Purple Crystals around the Portal.

Nessus:

Vex Conflux Defence - You need to fill the Circles with light which are around the Conflux, rather than killing enemies.

Blightmaker Attack - Stand in the smaller Bubbles of Blight, until you receive the Blight Receding Buff, then damage the Main Bubble of Blight while the buff is up.

Io:

Cabal Excavation - During the event, a Thresher Ship will fly around the map. Destroy the Thresher, as well as the Large Boss which will drop from the fallen Thresher.

Servitor