What is the most important source of natural gas? Coal, oil or other?
The most important source of natural gas is natural gas. This is why it is called natural. It is not "made" from oil or coal.

Natural gas forms by decomposition of organic material. Whether the organic material decomposes to coal, oil, or gas depends on the composition of the original material, time, pressure, and temperature. In theory, oil (=long chains of hydrocarbons) can decompose to gas (=mostly methane and small amounts of very short chains of hydrocarbons). I am not certain as to how much of the natural gas in existence went through this oil-like stage.

Coal cannot "become" natural hydrocarbon gas, because it contains no hydrogen. It can be burnt to form carbon dioxide, which is a gas, but this is not what you're talking about.