Sign of Connection

Bungie handed these emblems out during EGX 2018.

First to the Wild

Purchase anything on the Bungie Store during Season of the Drifter.

The Collector

This emblem dropped upon completion of the "Triumphant Collector" Triumph. No longer obtainable once Gambit Prime was retired.

Archivist's Prize

Purchase the "Destiny Grimoire Anthology, Volume II: Fallen Kingdoms" book. The emblem code will be inside the book.

Peace of the City

Purchase a Destiny 2-branded can of Rockstar Energy and use the promotional code. The emblem code should be sent to you upon using your first promotional code.

Carrhae

Purchase the "Destiny Ghost Vinyl - Generalist Shell" and the emblem code will come in the box. As of early 2021, these Ghost Shells don't seem to be readily available. You may still be able to find them second hand, but please make sure any ones you buy come with the emblem code.

Codex Doctrinae

Purchase the "Destiny Grimoire Anthology Volume I" and the emblem code will be inside the book.

Senior Recruiter

Complete your first Refer-a-Friend quest chain; this emblem will drop upon completion of the last step of the quest chain, completing a Nightfall strike with your linked friend.

Virescent

Earned as a rare drop in the Verdant Forest by defeating two or more bosses in a single run during the Season of the Drifter.

A Forest of Red

This emblem randomly dropped upon completion of this Nightfall. No longer obtainable since this strike has entered the DCV.

Rasmussen's Gift

Source: Solve the Norse glyph puzzle.

An Inscrutable Amygdaloid Eigenstate

This emblem randomly dropped upon completion of this Nightfall. No longer obtainable since this strike has entered the DCV.

Giantfall

This emblem randomly dropped upon completion of this Nightfall. No longer obtainable since this strike has entered the DCV.

Savior of the Past

In the Corridors of Time, follow this pattern: Diamond, Clover, Snake, Plus, Hex, Hex, Plus, Hex, Diamond, Clover, Snake (Credit to /r/raidsecrets).

Dance the Demons Away

Complete the Gambit Triumph "In Sync," where the entire team deposits 60 motes at once. This emblem will drop once you redeem the Triumph. Removed in Destiny 2 v3.0.0.0.

Buried City

Participate in the first Destiny 2 Community Challenge: Mars, which takes place between 2019-09-03 and 2019-09-10. The community completed the challenges and the emblem was released on 2019-09-27.

Praxic Fire

This emblem randomly dropped upon completion of this Nightfall. No longer obtainable since this strike has entered the DCV.

Wheel of Fortune

Complete a flawless run of Tier III Reckoning twice, once with each boss.

Ennead

Present A Small Gift to all nine (9) of the Dreaming City cats.

Arctic Arsenalist

Collect all BrayTech and Arctic weapons on Mars. No longer obtainable after Mars entered the DCV.