If you manage to find a research task that rewards you stardust upon completion you can receive anywhere from 100 – 1,500. ... They are boostable by star pieces, so if you happen to have a research that rewards a high stardust amount, you might want to hold off claiming it until you have some eggs about to hatch.

Do star pieces work on quests?

Star Piece does increase dust rewards from quests.

When should I use a star piece?

Like Lucky Eggs, it's wise to wait until the right time before using them. Waiting until you're about to hatch Eggs, get a 7 Day Daily Bonus, and go on a catching spree is ideal, as well as waiting for a double Stardust event to really maximize your returns.