Here's a sentence in question: The children went home to their house. Should "house" be singular or plural? What about if the children follow their nose? Is "nose" singular or plural?

The choice comes down to whether there's one or more of the object in question. The children aren't all following one nose, right? Each child has a nose to follow. So, the children follow their noses.

The other sentence, however, could go either way. If the children share a house and they're all going back to that same house, it should be singular:

The children went home to their house.

If they are going to different houses, it should be plural:

The children went home to their houses.

If the type of dwelling isn't significant, you could avoid this conundrum all together with a more direct sentence:

The children went home.