It's still early days of The Taken King so I'm not sure if I've just missed something, but, I still don't have my Sunsinger Subclass continuation quest. I finished both the Voidwalker and Sunsinger quests we could do the week before The Taken King launched.
Once I began playing The Taken King, I did get my Voidwalker one (Energy drain kills in the Crucible), I have not got the Sunsinger (or Stormcaller, but that may be because I've not fully upgraded Stormcaller yet?) subclass quest back.
I have tried switching to it in orbit and going to see Ikora, along with using Sunsinger in PvE and PvP and then going to check with her, but nothing has worked so far.
If there is information or fixs out there, I'd appreciate the knowledge.
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The new subclass on any character is unlocked by: 1. do the first mars mission 2. see the vanguards (plural) back at the tower, and one should give you the new quest for the new subclass 3. do the first one or two missions it asks for; sometimes I get auto directed into the second one after completing the first other times I have to choose it For old subclass quests, there are 2 very specific subclass quests, and there are separate crucible quests which are not related. Also, your quest screen has 2 or more pages, look for the arrows in the upper right corner, only 6 appear per screen but you can hold 32.