Fallen Glimmer Event
Destroy all three Glimmer nodes. All three must be destroyed. A secondary event will be triggered where you defend a point against waves of enemies.
Fallen Ether Event
Destroy all smaller servitors before eliminating the large one. The large servitor will be buffed and much harder to kill.
Spider Tank Event
Eliminate all three force fields using arc charges (two per field), which drop in sets of three when the weak point is exposed on the tank. A second tank will spawn once the heroic level is triggered.
Witches Ritual Event
Stand within the hex circles during the event and shoot the two crystals on either side of the portal. A Knight will spawn in addition to the Wizards.
Vex Construction Event
Active three plates by standing on them during the event. Many additional enemies will spawn in addition to a hydra.
Cabal Extraction Event
Begin to capture the drill. At 50 percent complete, a Cabal Thresher will spawn - destroy it. If it isn't destroyed before it leaves, it will spawn again around 75 percent. Once the Cabal Thresher is destroyed, a Cabal Excavator Unit will appear.
Taken Event
Stepping into and then out of the smaller blights will give you a 5 second buff called Blight Receding. The buff allows you to damage the main blight which when destroyed triggers the heroic version and spawns a large taken leader.
Cabal Injection Rig
The Cabal drop an enormous rig into an area, as you’re fighting but before you kill the Infiltrator Valus, look for the open heat vents on the rig. They’ll open in phases — three at the top, three in the middle, then one on the bottom — as you destroy them.
Once you trigger the Heroic Public Event, a Cabal Centurion will appear. He’s your new objective. Kill him to receive your big reward.
Mercury Vex event
Kill Vex and then destroy the four portals as normal. Once you are launched the first island off the side of the map destroy that portal as well but do not leave the island yet (as this can only be done at the first island you are sent to). At the back end of the island a prism will appearing float in the air. Destroy it and a platform will appear, do this two more times each time adding another prism to be destroyed. Once you get to the top of the island structure you just capture the Vex plate to start the Heroic version of the event.
Once the portal on the other island is destroyed, go to the middle of the area and damage the Gatelord as usual. At about 66% and 33% health, he will get an invulnerability shield. When this happens a Major Vex will spawn at one end of the middle area. Kill it, and dunk the orb in the area above the Gatelord (using the jump pads just outside the middle area makes this quick). This will bring the Gatelord's shield down and you can continue damaging him.
(Note that as of this writing, this public event is buggy: if you have triggered the Heroic version, but kill the Gatelord before his invulnerability shield is brought up twice, you will only get normal rewards for the event, not the Heroic rewards.)
Best Answer
I understand that the accepted answer implies that you will learn how to unlock these weirdly locked chests as you progress through the campaign; however, I'd still like to add a solid answer covered by a spoiler: