Data Mine

Access your first Sleeper Node on Mars. No longer obtainable after Mars entered the DCV.

Cold Comfort

Obtain your first BrayTech or Arctic weapon on Mars. The first Sleeper Node you open should give you an eligible weapon, also granting you this emblem at the same time. No longer obtainable since Mars entered the DCV.

Jacobson's Staff

From Destiny 2 v6.3.0 onward: Purchase from Failsafe on Nessus for 10 Legendary Shards.Previously: When you turn in enough planetary materials to rank up with Failsafe, she may very rarely drop this emblem when you open the engram she presents.

Specter of the Gun

Once you complete the Warmind campaign, this quest will become available on Mars. No longer obtainable since Mars entered the DCV.

IKELOS Imperative

Collect your first Seraph (AKA IKELOS) weapon. No longer obtainable after Mars entered the DCV.

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.