Nice attempt.
I'd be careful about using that Singleton. Apart from the fact that it is a Singleton with those inherent issues of shared mutable state, that implementation includes a great way to crash or have items error into oblivion during shutting down when you return null where callers are expecting an instance. Although Unity by default treats your game object code in a single thread, the lock pattern you have is rather contentious for multithreaded code, particularly if you have many threads attempting to read the object at once.
As for the pools, I'm not sure why you write that there are zero allocations. It looks like a pool Spawn can not only add to collections, but can also create entirely new stacks of objects. Adding to dictionaries is a potentially slow operation, so I'm not sure why you chose that container.
While it is a good attempt, I think I'd rather go with a flat container and a live/dead flag. And I'd recommend avoiding that Singleton, and avoiding shared mutable state in general.