Lore Scholar

You needed to obtain a minimum of 5,000 Grimoire score in Destiny 1 before August 1st, 2017.

Young Wolf

You needed to reach at least level 1 in the Ages of Triumph record book in Destiny 1 before August 1st, 2017. The Ages of Triumph record book was available as part of Rise of Iron.

Slayer of Oryx

You needed to complete one (1) Moment of Triumph during Destiny 1's second year, available as part of The Taken King.

Echo Mesa

This emblem originally dropped from Io reputation packages. No longer obtainable after Io entered the DCV.

The First Wave

This emblem is granted for participating in the Destiny 2 Beta.

Laurel Triumphant

Complete one (1) Moment of Triumph during Destiny 1's first year.

European Dead Zone

This emblem randomly drops from rank-up packages in the EDZ; it's a guaranteed drop after ten (10) packages. Appears to be vaulted after Beyond Light.

For Glory

This emblem was given out to players in celebration of Ranked PvP's release. Only tracks your stats from when it was released (at the beginning of Season 03) onwards.

For Valor

This emblem was given out to players in celebration of Ranked PvP's release. Only tracks your stats from when it was released (at the beginning of Season 03) onwards.

Rally the Flag

This emblem will drop upon your first Heroic public event completion. No longer drops after Beyond Light's release.

Crossroads Emblem

Obtained by completing a Heroic Public Event on Mercury. No longer obtainable after Mercury entered the DCV.

Playin' the Odds

Appears to be unlocked the first time you talk to the Drifter. If you have more concrete info, please contact me on Twitter @EmblemCollector.