The Wheel

Unknown source.

Royal Court

Unknown source.

Unsounded Depths

Complete the Sundered Doctrine dungeon for the first time.

Anomalous

Complete the Vesper's Host dungeon once.

Sunkissed

Complete the Solstice of Heroes 2023 Event Challenge, "Fuel for the Fire II." Emblem is obtained upon Challenge completion.

Victorious Oddment

Complete the Final Shape campaign on Legendary then redeem the Triumph.

Wander On

Obtained from your Ghost in The Lost City in the Pale Heart. Awarded in celebration of Guardians working to heal the Traveler. This emblem seems to drop at different times and after different quests for some Guardians. If you don't see it at your Ghost, complete more quests and it should eventually drop.

King No More

Clear the King's Fall raid for the first time. Obtained once the King's Fall completion Triumph is redeemed.

Sanctuary of Nefele

Otabined from completing the Lightfall Legendary campaign.

Wheel of Misfortune

Open your first Resplendant Reward Package from Xur in Eternity.

Shadow Market

Complete the Season 13 Umbral Engram tutorial, "Decrypting the Darkness"

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.

High-Risk, High-Reward

This emblem should unlock after finishing your first Nightfall strike.

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.