As for the evidence that abiogenesis took place, that is obvious. All models of the birth of the solar system involve conditions in which it would be impossible for earthly life to survive. But now, there is life on Earth. Ergo, if our models of the early Earth are good, inorganic chemistry at some point turned into biochemistry. That is all abiogenesis means.
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We can go further. The conditions in the *universe* at, say, 3 seconds into the current expansion, made it impossible for life to exist *anywhere*.
Now, life exists. So, at some point, there was a transition between non-life and life. That is abiogenesis.
The next line of evidence is quite firm: Life is a complex collection of chemical reactions that allow maintenance of internal state, growth, and reproduction.
In other words, there is nothing supernatural about life. There is no 'elan vitale', no 'spark of life', that distinguishes non-life from life. It should be pointed out that this is quite contrary to what people previously believed and goes against most religious dogmas about God 'breathing life' into.
A line of evidence related to this: there is no difference between a carbon atom in a living thing and a carbon atom in, say, carbon dioxide or a carbonate rock. In other words, life itself is made from non-living parts. So, instead of something magically becoming life, we realize that life is a matter of *organization* and not a matter of some new type of substance.
The next line of evidence is that the basic chemicals of life (amino acids, sugars, nucleic bases, etc) are either common in the universe or easily made from chemicals we know are common in the universe (ammonia, water, methane, carbon monoxide, carbon dioxide, etc). Again, at one time people thought the basic components of life would be difficult to make. This is where the Urey-Miller experiment and related experiments come in: in many different scenarios using what we *know* was present in the atmosphere of the early earth, the basic building blocks (amino acids, sugars, etc) are readily produced.
From the other direction: if all cells were as complicated as those in, say, a human body, then the issue of abiogenesis would be much harder. But, again, we know that eucaryotic cells like those in most multicellular organisms are 'collectives' of other, more primitive organisms. The most well known example is the mitochondria that are the energy centers of the cell. These are closely related to certain types of bacteria, even including their own DNA which has the bacterial structure (as opposed to using histones to wrap the DNA around).
So, we have strong evidence *from the study of biology* that the life we see today has evolved from single celled life. This, by the way, is exactly backwards from the creationists claims. The evidence for evolution doesn't relay on abiogenesis: all the evidence for evolution would still hold even if life originated supernaturally. We would still have the evidence linking single celled life to more complicated life even if life was originally produced supernaturally. This is what we mean when we say that evolution doens't depend on abiogenesis.
But what *is* true currently is that evolution is strong evidence for abiogenesis. We can even look into our modern cells and find remnants of the earliest forms of life. It was from the study of modern cells that the RNA world hypothesis originated. But this has reduced the issues surrounding the metabolism-first vs genetics-first arguments that previously surrounded the subject. Now, we know that RNA can do BOTH genetics and metabolism. It can even catalyze its own reproduction! In some definitions of the term 'life', certain strands of RNA are already alive.
So, we know that the basic chemicals of life are produced in large quantities in environments like that of the early Earth. Do we have evidence that these chemicals can spontaneously assemble into structures necessary for life? Yes! We know that they can, under a variety of conditions (again) produce cell-like structures that will bud off to produce new 'pseudocells' and that can catalyze reactions required for life (like glycolysis). The main reason these pseudocells are NOT considered to be alive is that the concentration of the chemicals inside tends to decrease with each budding.
So, do we have *evidence* of abiogenesis. YES. Plenty of it. Do we know specific mechanisms for how life actually arose on Earth? No. Because of the difficulty of figuring out the exact details of the environment of the early Earth, we may never know specifics.
But we do know that life is a complex collection of chemical reactions involving chemicals that are produced easily on the early Earth, that will spontaneously assemble into cell-lie structures that perform some basic functions of life and that life itself was considerably simpler in the past.