Destiny Emblem Collector is the most complete source for Destiny emblems: what they look like, exactly how to get them, and whether they're still available.
DEC currently tracks 499 Destiny 2 emblems and 271 Destiny 1 emblems.
Time-Limited Emblems
There are no time-limited emblems right now! Rest easy, Guardian.
Emblems Only Available During This Current Season
Season 12: Season of the Hunt runs from November 10th, 2020 to February 9th, 2021