I found my present flat completely by __.
What will be the most appropriate word in this sentence?
By chance is what you are looking for, as you found it unexpectedly. Also it sounds the most natural.
Actually, any of the words would work just about as well in your example: by chance and by coincidence emphasize the random nature of the occurrence and are interchangeable, while by luck adds an element of good fortune to it. The concept of luck mitigates the sense of randomness to some extent, as many people perceive luck to be a mystical, deterministic power that operates on behalf of someone.
So it all depends on what you mean.