Public events always give better rewards on Heroic — more Legendary engrams, more XP, and a shot at some event-exclusive drops. It's worth doing on any destination, but especially one that's currently Distorted, since that already bumps the gear tier on everything else you pick up out there.
Cosmodrome, EDZ, Nessus, Europa
Devil Walker
Shoot one of the Walker's legs to stagger it — that drops three Arc charges. Deposit those into the receivers powering the three blue shields nearby, then stagger it again for three more charges and deposit those too (six charges, six receivers total). That cracks open all three shielded Scorch Cannon caches and spawns a second Walker — kill both before time runs out. Do this before the first Walker dies, or the event ends without ever going Heroic.
EDZ
Injection Rig
Killing the Elite Psion guarding the Rig opens a set of vents for that tier — shoot just one of the three glowing vents to clear it (stand at the edge of the heat dome so the venting doesn't burn you). Repeat for the second and third tiers, one vent destroyed per tier, to trigger Heroic and bring in the boss, Infiltrator Valus.
EDZ, Nessus
Glimmer Extraction
A single Glimmer Refinery node spawns during each of the event's three phases. Destroy it every phase (three total, one per phase) to trigger Heroic, then stand in the resulting glimmer pool to capture it while the Fallen keep attacking.
EDZ
Cabal Excavation
Capture the drill like normal, but around 30-40% override progress a Cabal Thresher flies in for a couple of passes — shoot it down before it leaves. That spawns Excavation Valus, a Centurion you defeat to finish the Heroic event.
EDZ, Nessus, Moon, Europa
Ether Resupply
While you're working on the big Prime Ether Servitor, smaller Transport Servitors arrive in waves — one, then two, then three — and only stick around for 20-30 seconds before they leave. Destroy every single one, in every wave; missing even one means the event won't go Heroic. Clear them all and the Prime Ether Servitor enrages for a tougher, better-rewarding finish.
Nessus
Conflux
Follow one of the three faint white beams radiating out from the central conflux to an invisible capture plate — standing on it captures it, faster with more Guardians helping, while Vex Fanatics rush in to interrupt you. Capture all three plates and the event goes Heroic: the Vex throw everything at you, including Hydras, and you just need to survive until time runs out.
Europa
Crux Convergence
The event drops three Crux points, each guarded by a shielded Brig with three drones scanning it overhead (nine drones total). Stand in the circle under each Crux to fill its meter and drop that Brig's shield, then kill it — but clear every drone across all three points before you finish the third Brig, or you'll miss the Heroic window. Do it in time and a Fallen Commodore Brig drops in as an extra kill to finish the event.
Savathûn's Throne World
Witches' Ritual
Kill the Knight guarding each plate, then stand on the plates to strip the two Wizards' shields and kill them. That spawns two Hive crystals over the portal — stay on the plates to lower the crystals' shields too, then shoot both down quickly. That triggers Heroic and pulls a boss out of the portal to finish.
Tangled Shore — not currently in-game (vaulted)
Cryo-Pod Escape
Damage the escapee that pops out of the cryo-pod until it starts ground-slamming, which opens steaming vents on the pod — destroy those to reveal orbs, then throw the orbs at the escapee (twice per vent, within a short window) to freeze it instead of killing it. That triggers Heroic.