1) Logical evidence is limited in what it can prove regarding the existence of God. One reason is because of our limited experience with the material world and God.
Examples of logical arguments for God's existence are the ontological (Anselm of Canterbury and Descartes), cosmological (Aristotle and Thomas Aquinas) and teleological arguments. My favorite line of reasoning for God's existence is the Kalam cosmological argument. The Kalam argument says that everything which begins to exist has a cause; the universe began to exist; therefore, the universe has a cause. Now the cause of the universe is either personal or impersonal. But, based on what we know about how complex entities come into existence, it is plausible to believe that the universal cause of all things is personal.
2) Scientific arguments for God revolve around the universe's structure, its laws and constants. There are four basic physical forces that operate in the cosmos: gravity, electromagnetism, strong nuclear force and weak nuclear force. It has been observed that if electromagnetism were significantly weaker, then electrons would not be combine with atoms to form molecules. Furthermore, if the four basic forces were not fine-tuned as they appear to be, life as we know it would not be possible.
3) One argument that I've used recently with atheists has been the moral argument for God's existence, which was developed quite effectively by Immanuel Kant and suggested by Fyodor Dostoevski. The gist of this argument is that one must believe in God to make sense of morals or ethics. If God does not exist, or if we do not at least believe in God, then there is no foundation for absolute morality. It's also hard to make sense of similar morals that exist in diverse cultures, if God does not exist.