What kind of quests do you want to see in the next special research?

Quest for catching Celebi, something revolving around grass and psychic pokemon (evolving or catching them), and something that might involve time, such as "catch X amount of pokemon within this timeframe" (which could be done using an incense if insufficient spawns are around) or "walk this distance within this timeframe".

Catch a 100% Jirachi and use a super effective Doom Desire.

I don’t know if they made some tasks and independently made some rewards and then randomly assigned them but does it make any sense to get 1000 dust for evolving or catching something and also have a reward where you have to power up 5 somethings (minimum 1000 dust) to catch a single L15 starter (note field tasks, not special tasks)?

In that context, I think that your idea of “evolve all eeveelutions” is pretty good. The reward should be pretty awesome though, especially since it would require at least 20 km of walking for those who have already used the naming trick on espeon and umbreon. It would be a modest gate on the way to special. Easier for some who still have naming trick available...not unlike Magikarp evolve.

That said, I fully expect evolve swablu to be in the next round....and wailmer after that. Start saving.

I would like the normal catch screen to be repurposed into a battlefield where you have to choose one Pokemon to actually fight a wild Pokemon before catching it. That would be a cool task that I wouldn't ever delete. This would normally be Field Research tasks but it could implemented into Special Research too i.e ''Battle 10 Wild Pokemon in Field Research''