If I have Glass Mixologist alone at a battlefield and my opponent plays Singularity to kill it, can I play the summoned unit back to that same battlefield? What if I have another unit there?

Yes, you can play the unit from your trash to that battlefield in both scenarios.

The Ruling

According to the Glasc Mixologist card text, it has a Deathknell ability: "You may play a unit with cost no more than [3] and no more than [A] from your trash, ignoring its cost."

When Glasc Mixologist dies to Singularity, its Deathknell ability triggers. Because Deathknell is a triggered ability, it is placed on the chain as a pending item (Rule 322.3). When that ability resolves, you are instructed to play a unit from your trash.

Your Questions

1. If Glasc Mixologist is alone: Yes, you can play the unit to that battlefield. Even though the battlefield becomes unoccupied when Glasc Mixologist dies, the Deathknell trigger resolves after the unit has left play. You are permitted to play the new unit to that battlefield, which allows you to maintain control of it.
2. If you have another unit there: Yes, you can still play the unit to that battlefield. The presence of another unit simply means you never lost control of the battlefield in the first place.

Sequence of Events

1. Singularity is played and resolves, dealing 6 damage to Glasc Mixologist.
2. Glasc Mixologist dies and is sent to the trash.
3. The Deathknell ability is placed on the chain.
4. Players have an opportunity to play [REACTION] cards (if any).
5. The Deathknell ability resolves, allowing you to play a unit from your trash to the battlefield.

Note: This is my interpretation based on the mechanics of Deathknell and battlefield control. No official FAQ exists for this specific interaction, but it follows the standard resolution of triggered abilities during the cleanup/resolution process.