A. Under the window there stood a vase.
This is a presentational construction using a dummy there as Subject. The Locative Adjunct has been fronted. If we didn't have the fronting it would read:
- There stood a vase under the window.
B. Under the window stood a vase.
This is an example of Subject-Complement inversion. Here we see the Locative Complement of the verb STAND has swapped places with the indefinite Subject a vase. The sentence would otherwise read:
- A vase stood under the window.
C. Under the window there was a vase.
This is an existential construction in which the Locative Complement has been fronted. It would otherwise read:
- There was a vase under the window.
D. Under the window was a vase.
Again, this is an example of Subject-Complement inversion. This time with the verb BE instead of the verb STAND.
All four of these examples are completely grammatical. They all demonstrate strategies for avoiding using indefinite noun phrases as Subjects at the beginning of sentences. English speakers tend to avoid using indefinite noun phrases at the beginning of sentences where they can. This makes them easier for listeners to process.