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.