NB: This guide was originally written for live. I have been re-testing it lately on new characters, but I have not yet personally earned every Ouroboros badge on a single character, so there may still be a few surprises in there.
Ouroboros offers the Flashback system - a way for you to 'revisit' past missions, or missions you never played, changing 'your' history. And this system offers - you guessed it - badges. Lots and lots of badges.
This guide is based off of two threads archived from the boards: St. Angelius wrote the blueside one, and Nericus wrote the redside one. I wanted to have an updated copy that I could update as I experimented with the various Issue 25 updates. Then after a few years away, less than 20 minutes after I return I get a ping; so I'm updating this guide for Issue 26 and 27. Special thanks to BrandX, Burk, Rimy, and everyone else who commented with ideas for things to check out.
Unlocking Flashback
First, you need to become Entrusted With The Secret. To do so, once you're level 14, ask a player to open an Ouro portal for you (Pocket D is a busy place to ask, what with all the incarnate trials forming there), then climb to the top of the Ouroboros spire. Doing so will unlock the Ouroboros Portal power. You can also earn it accidentally a few other ways:
Complete Marshall Brass's 'Echo Down The Aeons' storyline (Cap au Diable, 15-19).
Complete Agent G's or Doc Delilah's story arcs (Faultline, 20-24).
During Vincent Ross's story arc (Sharkhead Isle, 20-29)
Complete the Ubelmann the Unknown story arc given by Ashwin Lannister or Laurie Pennington (Independence Port, 25-29).
Complete Holsten Armitage's mission (Brickstown, 30-40).
Complete Tina MacIntyre's Praetorian story arc (Peregrine Island, 40-44).
Complete any Patron story arc (45-50).
Complete the Barracuda Strike Force (Grandvale, 45-50).
Complete the Dr. Kahn (Founder's Falls, 45-50) or Ms. Liberty (Independence Port, 45-50) task forces.
Complete the final Kheldian story arc from Shadowstar or Sunstorm (45-50).
Earn most any badge in Recluse's Victory (Enter from Atlas Park City Hall, 50).
You now have access to the Pillar of Ice And Flame, the big magic supercomputer in the middle of the Ouroboros tower. This will allow you access to any mission that in 'your past' - which is to say, you've completed or outlevelled.
What Flashback badges are there?
Oh, so many. They're the same for heroes, villains, and Praetorians:
10 badges for number of Flashback arcs run; up to 50 runs total.
3 badges for speed of completion; one for each setting.
4 badges for limited lives; one for each setting.
8 badges for completing a story arc with debuffed players; one for each level range.
8 badges for completing a story arc with buffed enemies; one for each level range.
8 badges for completing a story arc without travel powers; one for each level range.
8 badges for completing a Flashback arc without temp owers; one for each level range.
2 badges for completing a Flashback without epics (ancillaries or patron pools); one for each level range you can have epics.
8 badges for completing a story arc without inspirations; one for each level range.
8 badges for completing a story arc without enhancements; one for each level range.
2 badges for completing a story arc without incarnate powers at level 50: one for alpha slot alone and one for without incarnate power at all.
1 badge for completing any Praetorian story arc.
This is in addition to the badges you can get from the missions themselves, which could be its own guide...
Note that the level ranges for the badges do not exactly match the current level ranges shown by the Aspect of the Pillar; these have shifted around somewhat, so things might not be exactly where you would find them if you played them in the "wild."
Optimizing runs
In the original boards, St. Angelius organized a four-run system: because the minimum number of times you have to play a story arc is four (one per power restriction, you repeat it four times, to get all the level range badges:
First, run the story arc with No Travel, Foes Buffed.
Second, run the story arc with Archetype Powers Only, No Inspirations.
Third, run the story arc with No Temporary Powers, Player Debuffed.
Finally, run the story arc with No Enhancements (and later, No Epic).
Each set combines a power challenge with the minor challenge least likely to interfere. With Player Debuffed and No Enhancements, you want all your inspirations and travel powers to stealth/avoid as much as you can - sneakiness is greatly rewarded by Ouroboros. The most powerful restriction is No Enhancements, which is coupled with the least-restricting power challenge, No Epics.
The challenges for lives and speed need only be done once (each), so do it when you've found a story arc you're comfortable with completing quickly. There are a few special restrictions for level 50, discussed in the level 50 section below.
Each run counts as a Task Force, with all the limitations that implies:
You can't change team halfway through. Players who drop out can't rejoin!
You can't do task forces or even talk to other contacts.
You do each story arc at the level of the story arc.
You can only exit by talking to the final mission NPC (or quitting the 'task force' in the Team menu of your chat bar).
Finally, many story arcs award reward merits, like any other task force.
Recommended Story Arcs
Here, I'll list things blueside and redside. Only objectives are spoiled below, no story content. Alternates are less optimal for various reasons, but are listed below in case the first option doesn't work for you or if you meet the specific requirements that makes the alternate a better choice.
Level 1-14
Hero: Officer Wincott Vs The Trolls (10-14, 0.02). 3 missions: defeat 10; defeat all, 4 glowies; defeat all, 3 glowies, boss.
Villain: The Loa Bone (1-9, 6.09). 1 mission: boss's mob. Bonus: Bone Collector badge.
Villain Alternate: The Hit List (10-14, 6.12). 3 missions: defeat boss; rescue NPC, escort NPC, glowie; 4 glowies, 1 boss.
Level 15-19
Hero: Vahzilok Wasting Disease (0.07). 1 timed mission: 3 glowies. Bonus: Plague Stopper badge.
Villain: Blood of Tyrants (Imperial Crusader, Ch 5, 18.31): 1 solo mission: talk, glowie, cutscene, moral choice (resistance: run to exit, you can run past mobs; loyalist: fight boss with allies)
Villain Alternate: Breakout At The Zig. 3 missions: click glowie; escort prisoners; defeat all.
Level 20-24
Hero: The Chameleon Suit (0.12). One mission: glowie. Bonus: Chameleon badge.
Hero alternate: I Lost My Daddy! (Faultline ch 2, 8.02). 5 missions: rescue NPC, escort NPC, ambush; talk to contact; defeat 10; find boss (beware mezzes); find NPC, escort NPC, ambush (beware mezzes).
Villain: Operation Skyhook. 4 missions: get glowie; talk to contact; kidnap NPC, escort NPC (timed!); 2 glowies.
Level 25-29
Hero: Black Channel Protocols (First War Ch. 4, 21.18): Defeat 3; escort (small map), talk; escort (small map), talk, cutscene, defend, boss; talk, boss, 2 glowies, boss. More combat but all in one neighborhood.
Hero alternate: The Redemption of Alexander "The Great" (0.23). 6 missions: talk to contact; patrol IP; talk to contact; defeat 35 (just north of IP north bus stop); kill all (medium map); talk to contact. Bonus: Redeemer badge. Theoretically less combat, but patrolling IP and going back and forth between a Talos contact and IP missions involves a lot of travel.
Villain: Eliminate Negotiator and Witnesses (6.45): 1 mission: two bosses. Bonus: Agent of Discord badge
Villain Alternate: On the Trail of the Leviathan. 3 missions: defeat 20; find glowie, ambushes; 5 glowies, boss.
Level 30-34
Hero: Defend the War Walls (0.28). 1 mission: rescue 8, defend 3, kill boss. Bonus: War Wall Defender badge. Note: Instead of starting from the pillar, use the Ouroboros sigil portal to board transit.
Hero alternate: Rise of the Vampyri. 10 missions: 10 glowies; glowie; defeat all, 2 glowies; 3 glowies; find correct glowies, defeat boss mob; defeat 10; defeat 15; defeat 20; click glowie, defeat boss mob; clear boss room.
Villain: The Unlucky Artifact (6.68). 1 mission: kill boss.
Villain Alternate: Breaking the Streak. 3 missions: Defeat 10; 2 glowies; defeat boss.
Level 35-39
Hero: They Call Me The Doctor (0.37): 1 mission: 2 keys, talk, 3 glowies. Bonus: Doctor's Ally badge.
Villain: Fat Cat City. 3 missions: Find glowie; find correct glowie; kill boss mob, escort NPC (ambushes!).
Level 40-49
Hero: The Clockwork King's Psychic Signal (1.09): 1 mission: defeat AV. Bonus: Emancipator Badge
Hero alternate: Division: Line. 13 missions: 6 glowies (timed!); 6 glowies; rescue NPC; rescue NPC; 4 rescue NPCs; glowie; glowie (ambush on exit!); rescue NPC, escort NPC; defeat boss mob; defeat boss mob; rescue NPC; 6 glowies; 10 glowies. Long but a lot is stealthable.
Villain: Absolute Vengeance. 3 missions: kill boss; clear boss room; kill boss mob.
Level 50
Level 50 is special because there are two badges you can only get at this level: Alpha Not Omega and Deincarnated. You can combine No Enhancements (the most powerful debuffs), with Alpha Only (the least restrictive power restriction) instead of No Epics and have an easier time.
Hero: Aura of Power: 5 missions: Talk, defeat 15 Freaks, defeat 15 Rikti, defeat 15 DE, (all in Crey's Folly by the Boomtown door), talk. Originally level 30, currently awarding level 50 badges.
Hero alternate: The Strange Case Of Benjamin A. Decker. 4 missions: rescue NPC, escort NPC, ambush; rescue NPC, guard NPC; talk to contact; 4 glowies (timed, NO combat allowed!).
Villain: Escaped! 3 missions: Kill boss; kill boss; kill boss.
Villain alternate: Emperor Cole's Personal Mission. 1 solo mission: Glowie, talk to NPC, cutscene, kill all with buffs. Bonus: Embodiment of Law badge. Easier than Escaped! but the Pillar of Ice and Flame doesn't know where to send you so you wind up doing more legwork. This is better if you have the Long Range Teleporter power, since that drops you right at the door for this mission.