God's intervention in the world happens spontaneously or sua sponte all the time (sua sponte means on God's own initiative and is a slightly better word than spontaneously which makes it seem like God just chaotically intervenes). I remember a training accident or two, where I was protected from harm and a seventy foot drop down a mountainside. Physical, tangible acts of God, (much like a friend of my father's who was blessed and had all of his tattoos disappear) except they happen without our asking.
Less tangible, but just as real, is inspiration. Most missionaries have encountered it. I've been on my way to an appointment and had a flash of light and clear directions. The appointment fell through and we slogged back through the snow, up the stairs and past the corner to the door, where a complete stranger was still hoping and hungering for us to come. Most can tell you of many such experiences.
I've had direct direction to speak to people or to help them when they had not thought to pray for help. It is easy to start to believe that inspiration and comfort are all that God does. I have friends who believe in what I call the "non contact" God -- a God who whispers into the world and who cares, but who lets every sparrow fall. It is easy to believe in a non-tangible God, but that vision would not be true. While comfort and guidance are important, God is more than just a comfort and a guide (or, as a friend said, happiness may be a compass and a warm blanket, but God is more).
Beyond intervention by God, sua sponte (there is that word again), there is also intercession (where God responds to prayer). Intervention is usually a delightful surprise, intercession is what we often plead for. We hunger for a God of miracles who can be interceded with.
I'll write on that next.