What are the major environmental drawbacks with using methane as a combustable fuel?
Methane has many lovely properties. It's a very dense source of energy. It is plentiful, it seemed to be cheap to extract (until we worked out the real cost), and similarly seemed to be cheap to transport long distances. It can be stored for many months in vast quantities.

But.

It's a greenhouse gas - a powerful one at that. On a 100-year horizon, it has about 20 times the global warming potential of $\ce{CO2}$. That needn't be a problem, if natural-gas extraction had no leakages, and distribution systems had no leakages.  But they do have leakages, which means the supply chain worsens global warming.

Furthermore, when methane is burnt, $\ce{CO2}$ is released into the atmosphere, worsening global warming even further.

We've learnt to handle methane's other main problems: that it's colourless and odourless, so can just accumulate unnoticed waiting to explode, if you let it; and that it's also dangerous if handled carelessly in any other way.

It also generates $NO_x$ when burnt in air, and they are local as well as global pollutants. It is possible to have flue-scrubbers to remove the $NO_x$.

That doesn't mean that that methane is permanently bad.  It can be created in a carbon-neutral fashion (from biological sources, or synthesised from atmospheric  $\ce{CO2}$), and if leakages are minimised, offers a potential useful form of energy storage. (note that in these forms, it would be an energy &amp; storage vector, not a net energy source, so its economic value would be very different - but that's economics, not chemistry)