I didn't fire the Blackstar, how can it best be exploited?

I found the Reaper Blackstar during Priority: Palaven and I resisted the urge to fire it, relying on conventional weapons to finish the mission.

It now appears next to my weapons bench in the Shuttle Bay.

What is the best way to use this single shot heavy weapon going forward? Is there a board where it will be highly useful or is there a plot twist that will benefit from it being present and unfired?

You don't.

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.

There's probably quite a few sequences where this could be useful - one that immediately comes to mind is the final fight in the mission on Lesuss, where you are investigating the Ardat-Yakshi monastery. This mission pops up after the events on the Krogan homeworld.

At the very end of this mission is a really rough fight against two very tough boss monsters. Being able to nuke one or both from orbit is probably a big bonus.

In response to Ben's position that carrying it would be painful - this mission is fairly short, cutscene heavy, and there's only a few combat situations to speak of, so carrying it through won't be too much of a burden.

My problem with these types of "one shot and it's gone" items (aka "Too Awesome To Use") is that I tend to hoard them for the last boss fight, and even then I'll hold off using them because I'm afraid the next final boss fight is going to be even worse.

My advice? Take it on this mission and use it. You won't be wasting it, and you won't have it laying on the weapon bench when the credits roll, doing you no good.