What is the best possible way to approach a woman that I fancy on the train without totally embarassing myself

Answer 1

Start by saying "hello". Especially to people who are regulars. Don't care if they answer back.

Once she gets used to you as a person who talks to her on the train (this may take a while), you can ask how they are, how their day was. All non-threatening, easy to back off if she shows no interest.

It's only if she becomes friendly and chats with you, that you should consider asking her out for coffee / drinks.

Answer 2

I have met three lovers on vehicles.

In one case, I looked pretty intently at a woman in a NYC subway station. I forgot about it, and took a seat on the train -- and she sat next to me.

In two other cases we were on a airplane, which of course leaves LOTS of time for conversation.

Basically, you just have to get close to her and talk. The choice of subject has more to do with what you feel comfortable with than what is "objectively" the best strategy. For example, I do better with an early joke, but that might not work for other people.