In my many debates with theists over the decades a handful of arguments for God’s existence are routinely articulated as “proofs” of divine providence. These include the cosmological argument (that all natural things are contingent on something else for their existence so there necessarily exists a being independent of nature), the ontological argument (that we can conceive of an absolutely perfect being means it must exist because existence is a necessary feature of perfection), the design argument (the universe is fine-tuned for life, and life contains design features, therefore God is the fine-tuner and intelligent designer of life), the moral argument (without God anything goes, with God there is objective morality), the consciousness argument (the qualitative experience—qualia—of consciousness cannot be explained by the activity of neurons, and abstract concepts like logic and mathematics exist separate from brains, therefore God must be the source), and others.
All of these arguments (they are certainly not proofs in the mathematical sense) have counter-arguments made by philosophers over the centuries, but there is one that seems to trouble a great many thinkers of all persuasions, and that is why there should be anything at all.