The wonder of evolution is that it works at all.
I mean that literally: If you want to marvel at evolution, that's what's marvel-worthy.
How does optimization first arise in the universe? If an intelligent agent designed Nature, who designed the intelligent agent? Where is the first design that has no designer? The puzzle is not how the first stage of the bootstrap can be super-clever and super-efficient; the puzzle is how it can happen at all.
Evolution resolves the infinite regression, not by being super-clever and super-efficient, but by being stupid and inefficient and working anyway. This is the marvel.
For professional reasons, I often have to discuss the slowness, randomness, and blindness of evolution. Afterward someone says: "You just said that evolution can't plan simultaneous changes, and that evolution is very inefficient because mutations are random. Isn't that what the creationists say? That you couldn't assemble a watch by randomly shaking the parts in a box?"
But the reply to creationists is not that you can assemble a watch by shaking the parts in a box. The reply is that this is not how evolution works. If you think that evolution does work by whirlwinds assembling 747s, then the creationists have successfully misrepresented biology to you; they've sold the strawman.
The real answer is that complex machinery evolves either incrementally, or by adapting previous complex machinery used for a new purpose. Squirrels jump from treetop to treetop using just their muscles, but the length they can jump depends to some extent on the aerodynamics of their bodies. So now there are flying squirrels, so aerodynamic they can glide short distances. If birds were wiped out, the descendants of flying squirrels might reoccupy that ecological niche in ten million years, gliding membranes transformed into wings. And the creationists would say, "What good is half a wing? You'd just fall down and splat. How could squirrelbirds possibly have evolved incrementally?"
That's how one complex adaptation can jump-start a new complex adaptation. Complexity can also accrete incrementally, starting from a single mutation.
First comes some gene A which is simple, but at least a little useful on its own, so that A increases to universality in the gene pool. Now along comes gene B, which is only useful in the presence of A, but A is reliably present in the gene pool, so there's a reliable selection pressure in favor of B. Now a modified version of A* arises, which depends on B, but doesn't break B's dependency on A/A*. Then along comes C, which depends on A* and B, and B*, which depends on A* and C. Soon you've got "irreducibly complex" machinery that breaks if you take out any single piece.
And yet you can still visualize the trail backward to that single piece: you can, without breaking the whole machine, make one piece less dependent on another piece, and do this a few times, until you can take out one whole piece without breaking the machine, and so on until you've turned a ticking watch back into a crude sundial.
Here's an example: DNA stores information very nicely, in a durable format that allows for exact duplication. A ribosome turns that stored information into a sequence of amino acids, a protein, which folds up into a variety of chemically active shapes. The combined system, DNA and ribosome, can build all sorts of protein machinery. But what good is DNA, without a ribosome that turns DNA information into proteins? What good is a ribosome, without DNA to tell it which proteins to make?
Organisms don't always leave fossils, and evolutionary biology can't always figure out the incremental pathway. But in this case we do know how it happened. RNA shares with DNA the property of being able to carry information and replicate itself, although RNA is less durable and copies less accurately. And RNA also shares the ability of proteins to fold up into chemically active shapes, though it's not as versatile as the amino acid chains of proteins. Almost certainly, RNA is the single A which predates the mutually dependent A* and B.
It's just as important to note that RNA does the combined job of DNA and proteins poorly, as that it does the combined job at all. It's amazing enough that a single molecule can both store information and manipulate chemistry. For it to do the job well would be a wholly unnecessary miracle.
What was the very first replicator ever to exist? It may well have been an RNA strand, because by some strange coincidence, the chemical ingredients of RNA are chemicals that would have arisen naturally on the prebiotic Earth of 4 billion years ago. Please note: evolution does not explain the origin of life; evolutionary biology is not supposed to explain the first replicator, because the first replicator does not come from another replicator. Evolution describes statistical trends in replication. The first replicator wasn't a statistical trend, it was a pure accident. The notion that evolution should explain the origin of life is a pure strawman—more creationist misrepresentation.
If you'd been watching the primordial soup on the day of the first replicator, the day that reshaped the Earth, you would not have been impressed by how well the first replicator replicated. The first replicator probably copied itself like a drunken monkey on LSD. It would have exhibited none of the signs of careful fine-tuning embodied in modern replicators, because the first replicator was an accident. It was not needful for that single strand of RNA, or chemical hypercycle, or pattern in clay, to replicate gracefully. It just had to happen at all. Even so, it was probably very improbable, considered in an isolated event—but it only had to happen once, and there were a lot of tide pools. A few billions of years later, the replicators are walking on the moon.
The first accidental replicator was the most important molecule in the history of time. But if you praised it too highly, attributing to it all sorts of wonderful replication-aiding capabilities, you would be missing the whole point.
Don't think that, in the political battle between evolutionists and creationists, whoever praises evolution must be on the side of science. Science has a very exact idea of the capabilities of evolution. If you praise evolution one millimeter higher than this, you're not "fighting on evolution's side" against creationism. You're being scientifically inaccurate, full stop. You're falling into a creationist trap by insisting that, yes, a whirlwind does have the power to assemble a 747! Isn't that amazing! How wonderfully intelligent is evolution, how praiseworthy! Look at me, I'm pledging my allegiance to science! The more nice things I say about evolution, the more I must be on evolution's side against the creationists!
But to praise evolution too highly destroys the real wonder, which is not how well evolution designs things, but that a naturally occurring process manages to design anything at all.
So let us dispose of the idea that evolution is a wonderful designer, or a wonderful conductor of species destinies, which we human beings ought to imitate. For human intelligence to imitate evolution as a designer, would be like a sophisticated modern bacterium trying to imitate the first replicator as a biochemist. As T. H. Huxley, "Darwin's Bulldog", put it:
Let us understand, once and for all, that the ethical progress of society depends, not on imitating the cosmic process, still less in running away from it, but in combating it.
Huxley didn't say that because he disbelieved in evolution, but because he understood it all too well.
Can you name any of these anomalies which "don't seem to make sense?"
There have been various evolutionary quandaries, where it's not clear how this or that organism evolved, but many of these have been resolved by further discoveries, which clarified the line of descent. There are some lineages that are still hazy, the evolution of bats for example, where our record of their lineage is poor because their bones are delicate and do not fossilize readily, but cases like these are not a source of confusion.
All the allegations I've heard of anomalies which supposedly do not make sense have been put forward by creationist or intelligent design proponents, and they were all based on some sort of misconception. There may be some outstanding sources of confusion in the field of evolutionary biology, but if you have any in mind, you'll have to clarify.
This is not intended to be a comprehensive review, but evidence against the theory of evolution could occur in such cases as
A: Minor: traits are found where it is unclear how they could have evolved by a natural process, but not highly implausible. These are found on occasion, but are generally resolved in time. Many traits where the evolutionary pathway was once obscure are now understood, so the evidence of any particular trait is weak.
B: Major: Organisms are found in the fossil record where it is highly implausible that they could have evolved from existing precursors. Fossil rabbits in the Precambrian fall into this category. Highly complex and unusual organisms with no apparent precursors, when we should expect their precursors to have left some sort of record, such as large organisms with radial symmetry and internal skeletons, would also constitute strong evidence.
C. Extremely major evidence. Events like seeing a monkey give birth to a human, or humans give birth to mutants with wings or the power to throw bolts of lightning. Any highly reliable observations of organisms developing major, complex adaptations in a single step. For all that many people with low levels of scientific literacy think that this is the sort of thing evolution entails, this is actually the kind of observation that would essentially falsify it outright.
As for whether "existence happened by chance or design," the meaning of this assertion is unclear. What do you mean by existence? Do you mean species as they currently exist? The phenomenon of life itself? The universe? The nature of the discussion will depend on what you actually mean, but I'll note that a dichotomy between "design" and "chance" is probably not an appropriate characterization. If you drop a rock, must the fact that it falls either be by design or chance? If something happens as a predictable, inevitable consequence of the rules regarding how things behave, it makes little sense to call it a consequence of chance.
Scientists tend to treat evolution as something that is essentially not in question because our evidence for it is overwhelmingly comprehensive. While there may be some instances of confusion, they are very minor in comparison with our existing body of evidence, so while further observations might alter our understanding of the process in some way, they do not threaten the entire edifice of evolutionary theory any more than the Pioneer Anomaly (which was in fact recently explained without any change to our understanding of physics) threatened to undermine the idea that gravity exists.