Heavy weapons are a massive pain in the butt to move around; if you want to use another weapon, you'll have to drop the Blackstar/any heavy weapon. Since you can ONLY use the blackstar against something big or it feels wasted, this means you'll constantly be dropping and picking up the blackstar every few steps in many missions.
The one place where it could come in handy with little trouble is during the mission Extract Turian Survivors. There's a Harvester here for your second enemy encounter, and it could be amusing to kill it in one shot. In this case you only fight two small groups of enemies before you use the Blackstar. Any other mission you'd be dropping it dozens of times. As a side note, there's actually another blackstar near the end of this mission, that lets you take out another Harvester. If you really want to keep one around, you could take this blackstar, it's hard to miss.
The Blackstar isn't that special, it does show up in some other levels (right before it's appropriate for use) and you can get them as limited use weapons in Multiplayer. Except for the case of multiplayer or when the story gives you one right before a tough monster, the Blackstar just isn't very useful.
Apparently the one special thing you can do with the Blackstar is team damage, you can use it to hurt your squad. So, shooting James in the face is probably the most rewarding thing you can do with it.