I think it would be helpful in the discussion on this thread if you would detail your own experiences with evangelism of Muslims. How have you approached it? What methods and arguments did you use? Did you have any success?
Right off I have to say I have encountered a different people group — not Muslims, but Hindus (mostly, with a number of Christians, and probably a few Muslims, among them). It is in the context of a contract I have been working alongside of a lot of H1B Indians.
They are easier than Muslims to deal with, because they are so spiritual they will consider worshiping anything. Introducing Christ is very easy in such a context. I just tell them that’s whom I worship and then talk about Him. I stress the blessings of Christ and always wish them blessings. That is generally a winning formula. Sometimes it amuses me that these people get a better idea of Christ than some churchgoers.
I’ve had discussions with missionaries who do evangelize Muslims, so I am talking about second hand experience. The classic way is to spend quite a long time making social acquaintance. Then discussions about theological matters begin. The bible advice is to give a reason for the hope that is within us, but with meekness (i.e. self-control: don’t go over the top right away) and respect.
Special crisis situations could hasten this by providing a platform for witness. Say an Islamic family finds itself in trouble — an accident, a robbery, an illness. A Christian offers prayer and there is an answer of miraculous dimensions. The Christian then explains why the prayer was granted. This could work very quickly.
When this is looked at from the standpoint that the Christian is there to sell blessings... the logic becomes almost absurdly easy. It is the patience and work involved in carrying that out, which is not so easy.